MATH 4.1: Introducing differentiation	PPLATO @
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1.1 Module introduction

Many of the concepts and laws that physicists have to study involve the rate of change of one quantity with respect to another. Motion, which involves ideas such as position, velocity and acceleration provides many important examples. The velocity of an object is defined by the rate of change of its position with respect to time. The acceleration of an object is similarly defined by the rate of change of velocity with respect to time. In fact, rates of change are of such general importance that they have spawned an important branch of mathematics – differential calculus. This subject, created by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716), allows the everyday notion of ‘rate of change’ to be made mathematically precise and has many applications throughout science and technology.

This module defines the important concept of the derivative and explains its role in evaluating rates of change. In Section 2 we adopt a graphical approach to rates of change and identify them with the gradients of tangents to graphs. This is done in the context of linear motion, introducing both velocity and acceleration as the gradients of appropriately constructed graphs. (The material in this section essentially repeats the discussion of linear motion in the physics strand of FLAP. If you have already studied that material you need only remind yourself of the terminology and the main results.)

Although the discussion in Section 2 is restricted to simple cases, this graphical approach is cumbersome and it is quite clear that a simple algebraic technique for evaluating gradients would be of great value. The mathematical process known as differentiation provides just such a technique, and it is here that the derivative enters the discussion. In Subsection 3.1 the link between functions and graphs (essentially the notion that a curve can be described by an equation) is used to introduce the idea of a derivative, while in Subsection 3.2Subsections 3.2 and Subsection 3.33.3 differentiation is introduced and used it to find some simple derivatives. A formal definition of the derivative is given in Subsection 3.4 and applied to velocity and acceleration in Subsection 3.5. Linear motion continues to provide the context and motivation throughout this discussion, but Subsection 3.6 goes on to show how derivatives can also be used in a wide variety of other circumstances.

Throughout this module the emphasis is on basic ideas and understanding rather than on complicated calculations. The main aim is that you should understand the mathematical approach to rates of change, be able to interpret derivatives such as $\dfrac{dx}{dt}$, $\dfrac{dv}{dt}$ and $\dfrac{dy}{dx}$ graphically, and evaluate them algebraically in straightforward situations. The full development of the techniques and applications of differentiation is left to other modules.

1.2 Fast track questions

1.3 Ready to study?

2 Rates of change: graphs and gradients

2.1 Position–time graphs and constant velocity

Modern science is based on the idea that the world can only be understood by first making careful observations and measurements. Graphs provide an important way to represent such measurements. Although in the real world many things move along a curved path, it is obviously simpler to start with linear motion, i.e. motion along a straight line. This turns out to be less restrictive than it might initially appear, since any motion not involving spinning objects can be expressed as a combination of movements along two or three straight lines.

Table 1 The position of a car at various times.

Position coordinate x/m	Time t/s	Position coordinate x/m	Time t/s
0	0.0	35	68.0
5	1.7	40	84.0
10	6.8	45	99.0
15	15.0	50	115.0
20	26.0	55	130.0
25	39.0	60	146.0
30	53.0

Figure 2 shows a car and a pedestrian moving along a straight line. The line has been designated as the x–axis of some appropriately chosen system of Cartesian coordinates which has its origin at the fixed point indicated in the figure. At any time t, the position of the car or the pedestrian is completely specified by its position coordinate x which determines its displacement from the origin at that time. Note that x is measured in metres and may be positive or negative according to which side of the origin a point is located. At the particular instant pictured in Figure 2 the car is located at the point x = 30 m and the pedestrian is at x = −20 m. i

The position of a point automatically determines another quantity – its distance from the origin. This is the magnitude of its displacement from the origin. Of course, distances have to be positive quantities (it makes no sense to speak of a distance of −20 m) so we define the distance from the origin to any point on the line as |x|, the modulus or absolute value of its position coordinate. You will recall that absolute values are always positive, so |−20 m| = 20 m.

Table 1 shows the position of the car at 5-second intervals, starting from the time when it passed through the origin.

The corresponding graph, called the position–time graph of the motion, is shown in Figure 3. i As you can see, after about 30 s the graph is a straight line (i.e. linear) indicating that the car’s position coordinate is increasing by equal amounts in equal intervals of time. In other words, the rate of change of the car’s position with respect to time is constant after about 30 s.

Now, for an object moving along the x–axis, the rate of change of its position coordinate x with respect to time defines its velocity, v_x, which tells us how fast it moves and in which direction. (If the object moves in such a way that x increases with time then v_x is positive; if it moves in the opposite direction so that x decreases with time then v_x is negative.) i So we can interpret the linear part of position–time graph as indicating that the car has a constant (or uniform) velocity after about 30 s.

We can summarize this discussion as follows:

For linear motion with uniform velocity, the position–time graph is linear, and the gradient of the position–time graph represents the rate of change of position with respect to time, i.e. the velocity.

An important point to notice about motion along the x–axis with uniform velocity v_x is that it implies that the moving object is always travelling in the same direction. Sometimes we may want to know how rapidly an object is moving but we may not care about its direction of motion. Under such circumstances the quantity likely to interest us is the speed of the object, given by

v = |v_x| = magnitude of velocity

Since |v_x| can never be negative it is clear that the speed v can never tell us anything about the direction of motion. Objects moving in opposite directions with velocities of 20 m s⁻¹ and −20 m s⁻¹ will both have a speed of 20 m s⁻¹.

2.2 Position–time graphs and instantaneous velocity

You saw in the last subsection that the gradient of a linear position– time graph can represent a constant velocity. This idea of using gradients is also the key to dealing with velocities that are not constant. Figure 6 is an enlargement of Figure 3 in the interval from 0 to 20 s. In this case the graph is curved which indicates that the velocity is changing. As you will see it is still true that the velocity at any particular time is given by the gradient of the graph at that time, but what exactly do we mean by the gradient at a particular time when the graph is curved? For example, in Figure 6, what is the gradient of the curve at t = 5 s? The definition of the gradient at a point on a curved graph occupied Newton and Leibniz, the founders of the calculus, more than a little and we will now consider that question in the context of position–time graphs.

How should we set about finding the velocity of the car at t = 5 s? Let’s try something similar to our approach in the last subsection.

In the 15 s interval from 5 s to 20 s the position coordinate changes by 24.3 m, from 1.7 m to 26.0 m. If we divide the change in position coordinate by the time interval we get the average velocity $\langle v_x\rangle$ over that time interval.

So, over the interval from 5 s to 20 s,

$\langle v_x\rangle$ = 24.3 m/15 s = 1.6 m s⁻¹

We can now measure the average velocity over a shorter interval starting at 5 s. For example, over the interval from 5 s to 15 s, the change in position coordinate is 13.3 m.

So, over the interval from 5 s to 15 s,

$\langle v_x\rangle$ = 13.3 m/10 s = 1.3 m s⁻¹

The values are plotted in Figure 7

In principle we could go on like this, using shorter and shorter intervals, but in practice it becomes impossible to continue when the interval becomes so short that the change in position coordinate cannot be reliably determined from the graph. Fortunately, we do not really need to do that as you are about to see.

Now imagine using intervals starting at 5 s that are much shorter than those in Question T3 to evaluate some more average velocities.

The velocity of 0.70 m s⁻¹ is called the limit of the average velocity, i.e. the value that is reached as the time interval from 5 s is made smaller and smaller. This limit gives us a definition of the instantaneous velocity at any particular moment:

instantaneous velocity at time t = the limit of the average velocity over an interval around t as that interval is made smaller and smaller

Note that by means of this limiting procedure we are able to make sense of the phrase ‘instantaneous velocity’ even though the moving object can’t really travel any distance at all in an ‘instant’. In general, ‘the velocity’ of an object is taken to refer to its instantaneous velocity at a particular time. Likewise, ‘the speed’ is generally taken to refer to the instantaneous speed, i.e. the magnitude of the (instantaneous) velocity.

It is useful to view what we have done in terms of the gradients of chords, i.e. straight lines that cut the curve at two points.

In Figure 8, the upper straight line passes through the points on the graph corresponding to the times t = 5 s and t = 20 s. The gradient of this straight line is the average velocity over that time interval. Other average velocities over shorter intervals are given by the gradients of the other lines in Figure 8, each of which cuts the curve at the beginning and endpoint of the relevant interval. As the intervals become smaller and smaller the lines crowd together and become nearly indistinguishable from one another. This corresponds to the average velocity becoming closer and closer to its limit.

Figure 9 shows the straight line that just touches the graph at 5 s. This line is the ‘limit’ of the lines drawn in Figure 8. It is called the tangent_to_a_curvetangent at t = 5 s, and its gradient gives the instantaneous velocity at t = 5 s.

Although limited in accuracy because of the difficulty of drawing tangents we now have an alternative definition of the instantaneous velocity. At any time t:

instantaneous velocity = gradient of the tangent to the position–time graph

Eventually, this approach will allow us to develop an algebraic technique which is much easier to apply than the graphical method, but before doing that we will consider how the ideas we have already developed allow us to deal with acceleration.

2.3 Velocity–time graphs and instantaneous acceleration

Table 2 shows the measured velocity v_x at six instants for a ball falling freely from rest. Just as Table 1 was used to plot a position–time graph, so we can use Table 2 to obtain the velocity–time graph shown in Figure 10.

In this case, the graph is a straight line, i.e. for any given interval of time the velocity changes by a constant amount. The rate of change of velocity with respect to time is called acceleration, and it can be found from the gradient of the velocity–time graph. i Figure 10 shows that for our freely–falling ball the acceleration in the first second is constant or uniform.

For linear motion with constant (or uniform) acceleration, the velocity–time graph is linear, and its gradient represents the rate of change of velocity with respect to time, i.e. the acceleration.

 

Table 2 Instantaneous velocity of a falling ball during
the first second after being released from rest.

Time from release t/s	Velocity of falling ball vx1/ m s−1
0.0	0.00
0.2	1.97
0.4	3.93
0.6	6.10
0.8	7.81
1.0	9.80

Acceleration, like velocity, has direction as well as magnitude. By giving the velocities in Table 2 a positive sign, we are saying that the downwards direction is positive and, since the velocities are increasing with time, the acceleration is also positive. If the velocities were decreasing in magnitude, but still positive, the acceleration would be negative.

It’s worth noting that a negative acceleration does not always slow things down, as the following question shows.

The term deceleration is often used to describe the slowing down of a body. It is sometimes stated that deceleration is negative acceleration but, as the question above shows, that is not necessarily the case – a negative acceleration only causes a reduction in speed if the motion is in the direction defined as positive.

Accurate measurement of the acceleration of freely–falling objects in the absence of air resistance gives a value of 9.8 m s⁻² (to two significant figures). However, our ball is not falling in a vacuum so air resistance i tends to slow its fall, as Figure 11 shows. After a few seconds the graph curves, rising less and less steeply. Hence the velocity is increasing less rapidly and the acceleration is no longer constant. (This is because the air resistance becomes greater as the ball’s velocity increases.)

Faced with this changing acceleration we can take the same approach to instantaneous acceleration as we took to instantaneous velocity: draw a tangent to the curve at the relevant time and measure its gradient. In Figure 11 a tangent has been drawn at the point on the curve where the time is 3.75 s. The tangent has a gradient of 3.9 m s⁻², so this is the instantaneous acceleration of the ball at 3.75 s. So, for linear motion, at any time t:

instantaneous acceleration = gradient of the tangent to the velocity–time graph

2.4 Gradients and rates of change in general

The crucial relationship in our discussion has been between the gradient of a tangent at a point on a curve and a rate of change. The gradient of the tangent at a point on the position–time curve gives the rate of change of position with respect to time, i.e. the instantaneous velocity, at the time corresponding to where the tangent is drawn. The gradient of the tangent at a point on the velocity–time curve gives the rate of change of velocity with respect to time, i.e. the instantaneous acceleration, again at the time corresponding to where the tangent is drawn. This approach works in many other situations. For example, if we have a graph showing how the temperature of an object changes with time then the gradient of the tangent at a point on that graph gives the instantaneous rate of change of temperature with respect to time at that specific time. The gradients of the tangents at different points on the curve give the rates of change of temperature at different times. Although the expression ‘rate of change’ is used, the quantity plotted horizontally need not be time. If temperature is plotted against altitude then drawing a tangent to the curve and finding its gradient will give the rate of change of temperature with respect to altitude at the altitude corresponding to the point where the tangent is drawn. Another example is the plot of the magnitude of the force exerted by a spring against its extension. Again the gradient at any point gives the rate of change of force magnitude with respect to extension at that point.

3 Rates of change: functions and derivatives

Section 2 should have convinced you that determining the gradients of tangents can provide useful information in many situations. However, performing this task graphically is very tedious and not always precise. Fortunately there is a non–graphical technique for calculating gradients that can be applied if we know the equation that describes the shape of a graph. This technique is called differentiation. It is part of that branch of mathematics known as calculus and was developed initially to study motion. In this section, we will briefly review the concept of a function and then introduce differentiation.

3.1 Functions, graphs and derivatives

Crudely speaking, a function is a rule (usually written as an algebraic equation) that relates two sets (usually sets of numbers or values). For example, the rule x = t² relates any real number t to a non–negative real number x. We show that x is determined by t by writing

x(t) = t²   for any real number t

and we say that x is a function of t. For any value of t we can use the given rule to work out a corresponding value of x; if t = 2 then x = 4, if t = 3 then x = 9 and so on. (Note that when we write x(t) in this context we do not mean x × t.)

Table 3 A table of values
for the function x(t) = t².

t	x (t)
0.0	0.00
0.5	0.25
1.0	1.00
1.5	2.25
2.0	4.00
2.5	6.25
3.0	9.00

When dealing with functions of this kind we call x and t variables and we say that x is a dependent variable, the value of which depends upon the value of the independent variable t.

Table 3 lists some representative values of x(t) for various values of t while Figure 14 shows the graph of the function obtained by plotting x(t) along the vertical axis and t along the horizontal axis. i

A note on mathematical terminology i The set of allowed values for the independent variable of a function is called the domain of the function. Another set called the codomain contains all of the allowed values of the dependent variable (and possibly some other values besides). The formal definition of a function requires that the rule which defines the function should be applicable to each value within its domain and that corresponding to each such value there should be a single value in the codomain.

It is easy to discuss linear motion in terms of functions. For instance, suppose an object is moving along the x–axis of a coordinate system with constant velocity u_x. If we let t represent the time that has elapsed since the object passed through the origin, we can say that the position x of the object is a function of the time t and is given by

x(t) = u_xt

Similarly, the position of an object that starts from rest at the origin at time t = 0 and moves with constant acceleration a_x is given by the function

x(t) = ½ a_xt² i

Note that the functions are different in these two cases because the physical circumstances (constant velocity in the first case, constant acceleration in the second) are different. In general we might expect that any particular kind of linear motion will correspond to a particular form for the function x(t).

Now, the process of differentiation has the following purpose:

Given a function x(t), i the process of differentiation can provide a related function x′(t) such that the value of x′(t) at any particular value of t is equal to the gradient of the tangent to the graph of x against t at that particular value of t.

The new function x′(t) produced by the process of differentiation is called the derived function or derivative of x(t) and is usually indicated by the symbol $\dfrac{dx}{dt}$ or, more formally,$\dfrac{dx}{dt}(t)$ to remind us that:

• its value at any particular value of t is equal to the gradient of the graph of x against t at that value of t.
• it is a function of t;

Note that $\dfrac{dx}{dt}$ is not a ratio of two quantities dx and dt, it is a single symbol that represents a particular function. This is why we sometimes write this as $\dfrac{dx}{dt}(t)$.

3.2 A linear function and its derivative

This subsection uses the algebraic view of a function to calculate derivatives and hence the gradients of graphs of functions. Soon you will be able to dispense with the graphs altogether and calculate derivatives directly.

The most general form of a linear function is x(t) = mt + c, where m and c are constants. The graph of such a function is a straight line of gradient m that intersects the vertical axis at c. Figure 15 shows the graph of x(t) = 0.5t + 2. The gradient of the graph at the point corresponding to t = 2 is 0.5, and it is clear that the gradient has the same value at any other point on the graph. In other words the derivative of x(t) = 0.5t + 2 is 0.5 at t = 2 and at every other value of t. So, in this case, x′(t) = dx/dt = 0.5 for all values of t.

In Figure 15, x and t represented numerical quantities, but if they had represented the position x (in metres) of an object at time t (in seconds), then the gradient of the graph would have represented the velocity v_x (in m s⁻¹) of the object. Thus the derivative $\dfrac{dx}{dt}(t)$ describes the velocity at time t.

We can summarize this result in Figure 16 and as follows:

The derivative of the linear function x(t) = mt + c is the constant function which takes the value m everywhere. This corresponds to the result that if the position coordinate, x, increases linearly with time, t, then the velocity v_x is constant.

3.3 A quadratic function and its derivative

Now that you have seen how to differentiate linear functions let us consider quadratic functions. Recall Figure 8 in which we used a sequence of chords i to approximate the tangent to a curve and thus found the gradient of the curve at a point. Let us apply this technique to the differentiation of a simple quadratic function.

Suppose the function we wish to differentiate is x(t) = t², and suppose we want to find the gradient when t = 3. The corresponding value of x is x(3) = 3². Imagine drawing a chord that cuts the graph of x(t) at t = 3 and cuts it again at another value which we denote t = 3 + ∆t (see Figure 17). i

Since x(t) = t² the value of x corresponding to t = 3 + ∆t is x = (3 + ∆t)². The gradient of the chord is therefore

$\text{gradient of chord} = \dfrac{\rm rise}{\rm run}$

$\phantom{\text{gradient of chord}} = \dfrac{(3+\Delta t)^2-3^2}{(3+\Delta t)-3} = \dfrac{9+6\Delta t+(\Delta t)^2-9}{\Delta t}$ i

$\phantom{\text{gradient of chord}} = \dfrac{6\Delta t +(\Delta t)^2}{\Delta t} = 6+\Delta t$

Let us use this expression to calculate the gradients of a sequence of different chords that provide closer and closer approximations to the tangent at t = 3. The chord which intersects the curve at t = 3 and t = 4 has ∆t = 1 and therefore a gradient of 6 + ∆t = 6 + 1 = 7. The gradients of the other chords with smaller ∆t approach the gradient of the tangent as ∆t approaches zero. But the gradient of each chord is given by 6 + ∆t, which approaches 6 as ∆t approaches zero. So 6 is the value for the gradient of the tangent at t = 3. You can demonstrate this for yourself in Question T8.

We can now generalize this process to obtain the gradient at any value of t. Pick a value t and a value close to it denoted by t + ∆t. The corresponding values of x are

x(t) = t²   and   x(t + ∆t) = (t + ∆t)²

The gradient of the chord that cuts the graph at t and t + ∆t is given by

${\rm gradient~of~chord} = \dfrac{x(t+\Delta t)-x(t)}{(t+\Delta t)-t} = \dfrac{(t+\Delta t)^2-t^2}{(t+\Delta t)-t}$

$\phantom{{\rm gradient~of~chord} }= \dfrac{t^2+2t(\Delta t)+(\Delta t)^2-t^2}{\Delta t} = 2t+\Delta t$

Now let the chord approach the tangent at t, i.e. let ∆t approach zero. This gives 2t as the gradient at time t. When t = 3 the gradient given by this expression is 6 (= 2 × 3) as we saw before. But we now have a general expression for the gradient at any value of t and hence an expression for the derived function $x^{\prime}(t) = \dfrac{dx}{dt}(t)$.

So, the derivative of a quadratic function   x(t) = t²   is   $\dfrac{dx}{dt}(t) = 2t$i

Note that the derivative dx/dt = 2t is zero when t = 0 and is negative when t is less than zero. This is exactly what we expect from the gradient of the graph of x(t) = t² – sketch the graph yourself if you are not convinced.

3.4 Formal definition of a derivative

In the linear and quadratic examples we have investigated so far, we found that at any value of t the derivative dx/dt of a function x(t) is approximated by the gradient ∆x/∆t of a chord. (In this expression ∆x = x(t + ∆t) − x(t) and this represents the change in x in the interval from t to t + ∆t.) We also found that the gradient ∆x/∆t provides an increasingly good approximation to dx/dt as ∆t approaches zero. Indeed, in the limit as ∆t goes to zero it will become exactly equal to the derivative of x(t) at t.

In general, given any function x(t) we define the derivative of x(t) by

Note that at any particular value of t, ∆x/∆t is a ratio of two quantities (∆x and ∆t), but dx/dt is not a ratio of dx and dt. Rather you should think of dx/dt at any particular value of t as the value of a function – the derived function x′(t) – at that value of t.

The definition of the derivative in Equation 1 is subject to the additional proviso ‘if a unique limit exists’. The proviso is necessary because there are some functions that cannot be differentiated at every point in their domain. i Sometimes $\dfrac{dx}{dt}(t)$ is written rather than dx/dt to stress that the derivative is a function.

A mathematical note A precise definition of the limit of an expression as ∆t approaches zero is surprisingly complicated. The limit is an expression that does not involve ∆t but which can be approached as closely as we wish by making ∆t small enough. The complication comes in giving precise meaning to the phrases ‘as closely as we wish ’ and ‘small enough’. In the cases we will meet the limit is obvious. For example, $\displaystyle \lim_{\Delta t\rightarrow 0} (6+\Delta t) = 6$ but there are cases where the limit is much less clear.

Although we are now thinking purely in terms of functions, rather than graphs, the process of finding a derivative looks much the same written down as it did before. For example,

ifx(t) = 4 − 2t + 3t²

thenx(t + ∆t) = 4 − 2(t + ∆t) + 3(t + ∆t)²

thus∆x = x(t + ∆t) − x(t) = 4 − 2t − 2(∆t) + 3t² + 6t(∆t) + 3(∆t)² − (4 − 2t + 3t²) = −2(∆t) + 6t(∆t) + 3(∆t)²

Hence∆x/∆t = − 2 + 6t + 3∆t

So$\displaystyle \dfrac{dx}{dt} = \lim_{\Delta t \rightarrow t}\left(\dfrac{\Delta x}{\Delta t}\right) = -2+6t$

3.5 Velocity and acceleration as derivatives

In this subsection we will gather together our results concerning derivatives and apply them to position, velocity and acceleration.

Suppose that x(t), v_x(t), a_x(t) are, respectively, the position, velocity and acceleration of an object at time t as it moves along the x–axis of a coordinate system. i Since velocity is the rate of change of position with respect to time we can write

Similarly, since acceleration is the rate of change of velocity with respect to time

Thus by two successive differentiations we can go from an expression giving the position to one giving the acceleration. The following questions illustrate this, and also yield some useful general results about differentiation.

In order to find the acceleration of this object we must differentiate again.

Since r is a constant we have found that a body whose position is a quadratic function of time must be moving with constant acceleration. In fact the converse also holds, and the position is a quadratic function if and only if the acceleration is a constant.

Finally, we note that when discussing linear motion it is a common practice to describe the location of the moving object in terms of its displacement s_x(t) from its initial position x(0). i Thus, rather than basing the discussion on the position x(t), the emphasis is on the difference in positions:

s_x(t)= x(t) − x(0)

This has the advantage that when t = 0 we are certain to find s_x(0) = 0. Since x(0) is a fixed point it is bound to be true that $\dfrac{ds_x}{dt} = \dfrac{dx}{dt}$, so at any time t the velocity of the moving object will be

$v_x(t)=\dfrac{ds_x}{dt}$ i

3.6 Derivatives and rates of change in general

We have seen that both velocity and acceleration are rates of change with respect to time. However ‘rate of change’ is a concept with wider application and does not have to be restricted to change with respect to time. For example, Figure 18 shows a spring aligned along the x–axis of a system of coordinates in such a way that when the spring is extended so that its free end is at x (where x ≥ 0) the force F_x (directed along the negative x–direction) exerted by the spring at its free end is F_x(x) = −kx where k is a constant. (In practice such an expression for F_x(x) might hold true over a limited range of values for x.) If you wanted to know how difficult it would be to increase the extension significantly you might well ask ‘what is the rate of change of the F_x with respect to x?’ In other words, ‘what is the derivative of F_x(x) with respect to x?’

dF_x/dx is calculated in the same way as dx/dt – only the letters have changed, not the underlying ideas. Since F_x(x) = −kx is a linear function of gradient k it follows that, dF_x/dx = −k. Note that in this case x is the independent variable and F_x is the dependent variable.

Although we have been using x to represent a position coordinate throughout much of this module there is no reason why it should always have that meaning. In fact it is common practice to use the letter x to represent a general independent variable, whilst a general dependent variable is commonly represented by y. (It is in this spirit that we refer to the horizontal and vertical axes of a graph as the x and y axes, respectively.) i

If we adopt this general notation and regard y as a function of x then, irrespective of the particular meaning we attach to x and y, we can define the rate of change of y with respect to x by

$\displaystyle \dfrac{dy}{dx} = \lim_{\Delta t \rightarrow 0}\left(\dfrac{\Delta y}{\Delta x}\right) = \lim_{\Delta t \rightarrow 0}\left[\dfrac{y(x+\Delta x)-y(x)}{\Delta x}\right]$(10) i

Remember that if a unique limit exists for all values of x in the domain of y(x) this procedure defines a function – the derived function y′(x) – which has the same domain as the original function.

4 Conclusion

4.1 The importance of differentiation

Much of physics is concerned with the dynamical processes involved with change and development. Differential calculus is the mathematics of change. It is hardly surprising therefore that many physicists would say that the development of calculus has been the single most important development in mathematics since the time of the ancient Greeks. Nor is it surprising that mastering the techniques of calculus is an important step in the development of every physicist.

In this module we have been mainly concerned with definitions and we have limited our considerations to the differentiation of linear and quadratic functions. In practice physicists rarely have to bother about limits and such mathematical niceties as domains and codomains; they rely on knowing the derivatives of some simple functions and a few basic rules that enable them to differentiate more complicated functions. Armed with this knowledge they are able to use differentiation to tackle an astonishing range of problems that would otherwise be quite intractable. Those wider skills and their applications are the subject of other FLAP modules.

5 Closing items

5.1 Module summary

1

In linear motion the position of a moving object can be specified by a single position coordinate x at any time t.

2

The (instantaneous) velocity v_x of an object in linear motion is the rate of change of its position coordinate with respect to time.

3

In linear motion, a plot of x against t is called a position–time graph. The gradient of the tangent to such a graph at any particular value of t determines the (instantaneous) velocity of the moving object at that time. The velocity is constant if and only if the position–time graph is a straight line.

4

The (instantaneous) acceleration a_x of an object in linear motion is the rate of change of its velocity with respect to time.

5

In linear motion, a plot of v_x against t is called a velocity–time graph. The gradient of the tangent to such a graph at any particular value of t determines the (instantaneous) acceleration of the moving object at that time. The acceleration is constant if and only if the velocity–time graph is a straight line.

6

If a single value of a dependent variable x can be associated with each value of an independent variable t in some specified domain, then we say that x is a function of t, and we speak of the function x(t) (pronounced ‘x of t’).

7

Given a function x(t), the rate of change of x with respect to t at any particular value of t is given by

$\displaystyle \dfrac{dx}{dt} = \lim_{\Delta t \rightarrow 0}\left(\dfrac{\Delta x}{\Delta t}\right) = \lim_{\Delta t \rightarrow 0}\left[\dfrac{x(t+\Delta t)-x(t)}{\Delta t}\right]$(Eqn 1)

If a unique limit exists for all values of t in some domain, then this formula defines a function called the derivative or derived function, written x′(t) or $\dfrac{dx}{dt}(t)$.

8

The value of the derivative dx/dt of a function x(t) at any given value of t is equal to the gradient of the tangent to the graph of x against t at that value of t.

9

In linear motion, the position coordinate of a moving object may be regarded as a function of time and written x(t). For such an object:

v_x(t) = dx/dt = rate of change of position coordinate with respect to time

a_x(t) = dv_x/dt = rate of change of velocity with respect to time

10

Generally, if y is a function of x, the derivative dy/dx describes the rate of change of y with respect to x.

If y(x) is the linear function y(x) = ax + b, then dy/dx = a

If y(x) is the quadratic function y(x) = ax² + bx + c, then dy/dx = 2ax + b

5.2 Achievements

Having completed this module, you should be able to:

A1

Define the terms that are emboldened and flagged in the margins of the module.

A2

Draw position–time and velocity–time graphs.

A3

Understand the relationship between rates of change and gradients of tangents and how they can be approximated by gradients of chords.

A4

Interpret the gradient of a tangent to a position–time or a velocity–time graph as velocity or acceleration, respectively.

A5

Recognize a linear position–time graph as describing constant velocity and a linear velocity–time graph as describing constant acceleration.

A6

Know the definition of a derivative as a limit and give a graphical interpretation of this definiton.

A7

Use the definition of a derivative as a limit to calculate the derivatives of linear and quadratic functions.

A8

Write down an expression for the derivative of a given linear or quadratic function.

A9

Express velocity and acceleration as derivatives for the case of linear motion and solve simple linear motion problems.

A10

Interpret rates of change in terms of the derivative of one quantity with respect to another.

Study comment You may now wish to take the following Exit test for this module which tests these Achievements. If you prefer to study the module further before taking this test then return to the topModule contents to review some of the topics.

5.3 Exit test

Study comment Having completed this module, you should be able to answer the following questions, each of which tests one or more of the Achievements.

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