Next exams: Wednesday 2nd May, Thursday 3rd May, Wednesday 9th May, Thursday 17th May, Friday 18th May

To differentiate composite functions of the form f(g(x)) we use the chain rule (or "function of a function" rule). The derivative of the function of a function f(g(x)) can be expressed as:
f'(g(x)).g'(x)

Alternatively if y=f(u) and u = g(x) then \displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

Alternatively if y=f(u) and u = g(x) then \displaystyle \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

## Summary/Background

The chain
rule is also known as the "function of a function" rule. It can be stated in a
number of ways.

If y = f(u), u = g(x) then \displaystyle\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}

Here are some more examples:

\displaystyle \frac{d((x+1)^6)}{dx} = 6(x+1)^5.1 = 6(x+1)^5

\displaystyle \frac{d(e^{10x})}{dx} = e^{10x}.10 = 10e^{10x}

\displaystyle \frac{d( \ln x^2)}{dx} = \frac{1}{x^2}.2x = \frac{2}{x}

If y = f(u), u = g(x) then \displaystyle\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}

Here are some more examples:

\displaystyle \frac{d((x+1)^6)}{dx} = 6(x+1)^5.1 = 6(x+1)^5

\displaystyle \frac{d(e^{10x})}{dx} = e^{10x}.10 = 10e^{10x}

\displaystyle \frac{d( \ln x^2)}{dx} = \frac{1}{x^2}.2x = \frac{2}{x}

## Software/Applets used on this page

## Glossary

### chain rule

A rule for differentiating a function of a function:

dy/dx = dy/du x du/dx.

dy/dx = dy/du x du/dx.

### composite

made of a combination of simpler shapes or bodies

### derivative

rate of change, dy/dx, f'(x), , Dx.

### differentiate

to find the derivative of a function

### function

A rule that connects one value in one set with one and only one value in another set.

### rule

A method for connecting one value with another.

### union

The union of two sets A and B is the set containing all the elements of A and B.