As you have learnt in the section on functions, a straight line graph has a gradient. This gradient describes the rate at which the graph is changing and thanks to it we can tell how steep the line will be. In fact gradients can be found for any function – the special thing about linear functions is that their gradient is always the same (given by m in y = mx + c ). For polynomial functions the gradient is always changing. This is where calculus comes in handy; we can use differentiation to derive a function using which we can find the gradient for any value of x.

Using the following steps, you can find the derivative function (f ′(x)) for any polynomial function (f(x)).

Polynomial a mathematical expression or function that contains several terms often raised to different powers

  e.g. y = 3x2, y = 121x5 + 7x3 + x or y = 4x + 2x


y = f(x) = axndy/dx = f′(x) = naxn−1.

The (original) function is described by y or f(x), the derivative (gradient) function is described by dy/dx or f ′(x).

Derivative of a constant (number)  0

e.g. For f(x) = 5, f′(x) = 0

Derivative of a sum sum of derivatives.

If a function you are looking to differentiate is made up of several summed parts, find the derivatives for each part separately and then add them together again.

e.g. f(x) = axn and g(x) = bxm
f′( x ) + g′( x ) = naxn−1 + mbxm−1

4.1.1 Rules

With more complicated functions, in which several functions are being multiplied or divided by one another (rather than just added or subtracted), you will need to use the product or quotient rules.

Product rule

Product rule


Product rule

Quotient rule

Quotient rule


Quotient rule

Chain Rule

When a function is inside another function: y = g(u) where u = f(x)

chain rule

Note: A function inside another function is a composite function, f ◦ g(x), which we discussed in the Functions chapter

Differentiating with the chain rule

Let y = (cos x)2, determine the derivative y′
  1. Determine what the inside (u) and outside (y) functions are

Inside function: u = cosx

Outside function: y = u2

2. Find u′ and y′

Chain rule

3. Fill in chain rule formula

chain rule