# Chain rule

In differential calculus, the chain rule is a way of finding the derivative of a function. It is used where the function is within another function. This is called a composite function.

More specifically, if ${\displaystyle F(x)}$ equals the composite function of the form:

${\displaystyle F(x)=f(g(x))}$

where g is a function differentiable at x and f is a function differentiable at g(x), then the derivative of ${\displaystyle F(x)}$, written as ${\displaystyle F'(x)}$, exists, and is equal to

${\displaystyle F'(x)=f'(g(x))g'(x)}$.[1][2][3]

## 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.

For example,

${\displaystyle F(x)=(x^{2}+5)^{3}\rightarrow 3(x^{2}+5)^{2}}$
${\displaystyle F'(x)=3(x^{2}+5)^{2}(2x)=6x(x^{2}+5)^{2}}$

In this example, the cubed sign (3) is the outside function and ${\displaystyle x^{2}+5}$ is the inside function. The derivative of the outside function would be ${\displaystyle 3x^{2}}$, where x is replaced by the inside function. The derivative of the inside function would be 2x, which is multiplied by ${\displaystyle 3(x^{2}+5)^{2}}$ to get ${\displaystyle 6x(x^{2}+5)^{2}}$.

## Proof

The very definition of a derivative is ${\displaystyle F'(x)=\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}}$.

With this knowledge:

${\displaystyle \lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}=\lim _{\Delta x\to 0}{\frac {\Delta y}{\cancel {\Delta u}}}\cdot {\frac {\cancel {\Delta u}}{\Delta x}}}$ must be true, which can also be written as ${\displaystyle F'(x)=f'(u(x))u(x)}$.[4]

## References

1. "Chain Rule for Derivative — The Theory". Math Vault. 2016-06-05. Retrieved 2020-09-19.
2. "Chain rule (article)". Khan Academy. Retrieved 2020-09-19.
3. Kouba, Duane (May 6, 1997). "Differentiation Using the Chain Rule". math.ucdavis.edu. Retrieved September 19, 2020.
4. "Chain rule proof (video) | Optional videos". Khan Academy. Retrieved 2022-09-11.