Differentiation

Derivatives

The derivative (or gradient function) of a function , is another function, that gives the gradient of the graph of at any point with x-coordinate . The gradient at any point of is the gradient of the tangent at that point.

The derivative itself can be graphed:

The relationship between the graph of a function and its derivative are:

When the graph is increasing, the gradient is positive.
When the graph is decreasing, the gradient is negative.
When the tangent is horizontal, the gradient is zero. This is a stationary point.

Differentiation from first principles

A chord is a line segment between two points on a curve, e.g. PQ above.
The idea behind differentiation from first principles is to consider the gradient of a chord, as the chord becomes smaller and smaller to find the gradient of the function at that point. This gives the (given) formula:

Second derivatives

The derivative has has two interpretations:

It is the gradient of the graph of against .
It measures the rate of change of with respect to .

Therefore, to calculate the gradient (or rate of change) at any particular point, substitute the value of into the equation for the derivative.

Increasing and decreasing functions

The sign of the gradient at a point shows whether the function is increasing or decreasing:

If the gradient is positive, the function is increasing.
If the gradient is negative, the function is decreasing.

Higher derivatives

The derivative can be differentiated with respect to . This gives the second derivative, denoted or . The second derivative measures the gradient of the gradient, or the rate of change of the gradient with respect to .

AS differentiation rules

For a general function :

If , where is a constant:

If :

These rules show how to differentiate powers, how to differentiate a function multiplied by a constant, and how to differentiate a sum.

Because of the power rule for the derivative of , it is useful to rearrange function into the form before differentiating them, by applying the laws of indices.

Tangents and normals

The normal to a curve at a given point is a line that crosses the curve at that point and is perpendicular to the tangent:

The gradient of the normal can be found by considering that for two perpendicular lines with gradients and , .

For a point on :

The gradient of the tangent is ,
The gradient of the normal is

Stationary points

At local maxima and minima, . These are local because it is possible that there are several points that have a gradient of zero - these are all stationary points.

To determine the nature of the stationary points, the second derivative can be used:

Given a stationary point:

If at that point, then it is a local maximum.
If at that point, then it is a local minimum.
If at that point, then a conclusion cannot be reached.

Concavity and points of inflection

Concave and convex describe functions that bend upwards or downwards:

A function is concave on an interval if in that interval.
A function is convex on an interval if in that interval.

A (semi-useful) mnemonic to remember this is that 'a caveman lives in a concave cave'.

A point of inflection occurs when and the sign of changes either side, i.e. goes from being concave to convex, or vice versa. A point of inflection can be stationary (if ), or non-stationary (otherwise).

A2 differentiation rules

The chain rule is used to differentiate composite functions (it is not given):

The product rule is used to differentiate products of functions (it is not given):

It can be remembered as 'left d-right plus right d-left'.

The quotient rule is used to differentiate quotients of functions (it is given):

The below derivatives are expected to be known:

While the remaining trig derivatives are given:

Examples

Below are examples for the A2 differentiation rules:

Connected rates of change

For connected rates of change, the following relations are used: