# Chain rule

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The **chain rule** is a way of finding the derivative of a function. It is used where the function is in another function. This is called a composite function.

If F(x) equals two functions that we can take a derivative of, such as:

- [math]F(x)=f(g(x))[/math]

then the derivative, F prime, is

- [math]F'(x)=f'(g(x))g'(x)[/math]

## Steps

**1.** Find the derivative of the outside function (all of it at once).

**2.** Find the derivative of the inside function (the bit between the brackets).

**3.** Multiply the answer from the first step by the answer from the second step. This is basically the last step in solving for the derivative of a function.

- Example;

- [math]F(x)=(x^2+5)^3[/math]
- [math]F(x)=3(x^2+5)^2[/math]
- [math]F'(x)=3(x^2+5)^2(2x)[/math]
- [math]F'(x)=6x(x^2+5)^2[/math]

In this example, the cubed sign (^{3}) is the outside function and [math]x^2+5[/math] is the inside function. The derivative of the outside function would be [math]3x^2[/math], where the inside function is plugged in for x. The derivative of the inside function would be 2x, which is multiplied by [math]3(x^2+5)^2[/math] to get [math]6x(x^2+5)^2[/math].