We will often need to differentiate functions that are more complex than the ones that we can already do. They will simply be variations, where 'x' has been replaced by a function 'f(x)'.

Example:

Differentiate the following:

1. y = (3x − 2)⁴
2. y = sin 5x
3. y = 2e^{(2x + 1)}
4. y = cos 4x

In each of these cases we can use a substitution to turn the expression into something we can differentiate.

Answer to 1:

We know how to differentiate x⁴, so we use the substitution u = (3x − 2) to turn the function into something that we can differentiate. This gives:

y = (3x − 2)⁴

Let u = 3x − 2 to give us,

y = u⁴,

Now differentiate to get:

The only problem is that we want dy/dx, not dy/du, and this is where we use the chain rule.

The chain rule says that

So all we need to do is to multiply dy/du by du/dx.

As u = 3x − 2, du/dx = 3, so

Answer to 2:

Differentiate y = sin 5x.

Let u = 5x (therefore, y = sin u)

so using the chain rule

So when using the chain rule:

1. Express the original function as a simpler function of u, where u is a function of x.
2.  
Differentiate the two functions you now have.
3.  
Multiply the derivatives together, leaving your answer in terms of the original question (i.e. in x's rather than u's).

For 3 and 4, see if you can do the workings and then check your answers against these:

Answer to 3:

Answer to 4:

When familiar with the chain rule, it is possible to produce a correct answer instantly without having to write down all the substitution working; simply follow through the three steps together.

Example:

Differentiate ln(x² + 3x + 3)

The denominator is from dy/du = 1/u, the numerator is du/dx)

In each of these formulae we have used the substitution u = f(x) and so the f ′ (x) corresponds to

(Have a go at using the chain rule to make the rules yourself.)