C1: Differentiation from First Principles

Learning Objective: to understand that differentiation is the process for

calculating the gradient of a curve. The rate of change can be calculated from first principles by considering the limit

of the function at any one point.

Rates of changeThis graph shows the distance that a car travels over a period of 5 seconds.

The gradient of the graph tells us the rate at which the distance changes with respect to time.

In other words, the gradient tells us the speed of the car.

change in distancegradient = =

change in time40

=5

8 m/s

The car in this example is travelling at a constant speed since the gradient is the same at every point on the graph.

time (s)

dist

ance

(m

)

0 5

40

Rates of changeIn most situations, however, the speed will not be constant and the distance–time graph will be curved.

For example, this graph shows the distance–time graph as the car moves off from rest.

The speed of the car, and therefore the gradient, changes as you move along the curve.

To find the rate of change in speed we need to find the gradient of the curve.

The process of finding the rate at which one variable changes with respect to another is called differentiation.

time (s)

dist

ance

(m

)

0

In most situations this involves finding the gradient of a curve.

The gradient of a curveThe gradient of a curve at a point is given by

the gradient of the tangent at that point.The gradient of a curve at a point is given by

the gradient of the tangent at that point.

Look at how the gradient changes as we move along a curve:

Differentiation from first principlesSuppose we want to find the gradient of a curve at a point A.

We can add another point B on the line close to point A.

As point B moves closer to point A, the gradient of the chord AB gets closer to the gradient of the tangent at A.

δx represents a small change in x and δy represents a small change in y.

Differentiation from first principlesWe can write the gradient of the chord AB as:

change in =

change in

y

x

y

x

As B gets closer to A, δx gets closer to 0 and gets closery

x

to the value of the gradient of the tangent at A.

δx can’t actually be equal to 0 because we would then have division by 0 and the gradient would then be undefined.

Instead we must consider the limit as δx tends to 0.

This means that δx becomes infinitesimal without actually becoming 0.

Differentiation from first principlesIf A is the point (3, 9) on the curve y = x2 and B is another point close to (3, 9) on the curve, we can write the coordinates of B as (3 + δx, (3 + δx)2).

The gradient of chord AB is:2(3 ) 9

(3 ) 3

x

x

δx

δy

29 6 ( ) 9=

x x

x

26 ( )=

x x

x

(6 )=

x x

x

=y

x

= 6 + x

A(3, 9)

B(3 + δx, (3 + δx)2)

Differentiation from first principles

We write this as:

Let’s apply this method to a general point on the curve y = x2.

If we let the x-coordinate of a general point A on the curve y = x2 be x, then the y-coordinate will by x2.

At the limit where δx → 0, 6 + δx → 6.

0lim =x

y

x

0

lim 6 + =x

x

6

So the gradient of the tangent to the curve y = x2 at the point (3, 9) is 6.

So, A is the point (x, x2).

If B is another point close to A(x, x2) on the curve, we can write the coordinates of B as (x + δx, (x + δx)2).

Differentiation from first principlesThe gradient of chord AB is:

2 2( )=

( )

y x x x

x x x x

δx

δy

2 2 22 ( )=

x x x x x

x

22 ( )=

x x x

x

(2 )=

x x x

x

2

0So for = , lim = 2

x

yy x x

x

= 2 +x x

A(x, x2)

B(x + δx, (x + δx)2)

The gradient functionSo the gradient of the tangent to the curve y = x2 at the general point (x, y) is 2x.

2x is often called the gradient function or the derived function of y = x2.

If the curve is written using function notation as y = f(x), then the derived function can be written as f ′(x).

So, if: f(x) = x2

This notation is useful if we want to find the gradient of f(x) at a particular point.

For example, the gradient of f(x) = x2 at the point (5, 25) is:

f ′(5) = 2 × 5 = 10

f ′(x) = 2xThen:

Now we shall differentiate y = x3 from first principles:

Using the notationWe have shown that for y = x3

dydx

represents the derivative of y with respect to x.dy

dx

2

0lim = 3x

yx

x

0limx

y

x

is usually written as .dy

dx

So if y = x3

of the tangent.

Remember, is the gradient of a chord, while is the gradienty

x

dy

dx

2= 3dy

xdx

then:

Using the notationdydx

This notation can be adapted for other variables so, for example:

Also, if we want to differentiate 2x4 with respect to x, for example, we can write:

represents the derivative of s with respect to t.ds

dt

We could work this out by differentiating from first principles, but in practice this is unusual.

4(2 )d

xdx

If s is distance and t is time then we can interpret this as the rate of change in distance with respect to time. In other words, the speed.

Task 1

Differentiate from first principles

1. y = x4

2. y = 1/x