Let's look into differentiation and the different rules in A-Level Maths!

**Chain rule:**

The chain rule is used to differentiate composite functions *y=f(g(x)) *(i.e. functions of a function)

The chain rule operates by making the substitution, *u=g(x). *Then

__Example__: Differentiate

**Product rule**

It is used when there is a product of 2 functions.

Example:

Differentiate y= (3x^{2} - 2x)e^{1-2x}

This is a product of 2 functions:

u = 3x^{2 }- 2x ⇨ du/dx = 6x - 2

v= e^{1-2x} ⇨ dv/dx= - 2e^{1-2x}

Using the productrule formula:

**Quotient rule**

It is used for differentiating quotients (i.e.fractions).

__Example__: Differentiate

Factorise top in order to simplify:

Remember these results:

**Finding the derivative when x = f(y):**

We use the result

Example:

If *x = 4y - y*^{2}, find *dy/dx* in terms of y.

**Connected rates of change:**

The word rate is associated with differentiation.

Key steps:

(1) Interpret mathematically the information given in the question.

(2) Write down the relevant formula connecting the variables and differentiate it.

(3) Decide on what quantity needs to be calculated.

(4) Use the chain rule to calculate it.

Example:

The volume of a sphere is decreasing at a rate of 20cm^{3}/s. Find the rate at which the radius is increasing when the radius is 3 cm.

Solution:

From question, *dV/dt = -20*

Connecting formula:

We want *dr/dt*

using chain rule

Drafted by Eunice (Maths)