Chain rule

Template:Multiple issues 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]\displaystyle{ F(x)=f(g(x)) }[/math]

then the derivative, F prime, is

[math]\displaystyle{ 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]\displaystyle{ F(x)=(x^2+5)^3 }[/math]

[math]\displaystyle{ F(x)=3(x^2+5)^2 }[/math]

[math]\displaystyle{ F'(x)=3(x^2+5)^2(2x) }[/math]

[math]\displaystyle{ F'(x)=6x(x^2+5)^2 }[/math]

In this example, the cubed sign (³) is the outside function and [math]\displaystyle{ x^2+5 }[/math] is the inside function. The derivative of the outside function would be [math]\displaystyle{ 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]\displaystyle{ 3(x^2+5)^2 }[/math] to get [math]\displaystyle{ 6x(x^2+5)^2 }[/math].