Go to content
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}

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}

You can get a better display of the maths by downloading special TeX fonts from jsMath. In the meantime, we will do the best we can with the fonts you have, but it may not be pretty and some equations may not be rendered correctly.

Glossary

chain rule

A rule for differentiating a function of a function:
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.

Full Glossary List