# Chain rule

The chain rule is a method used to determine the derivative of a composite function, where a composite function is a function comprised of a function of a function, such as f[g(x)].

Given that y(x) is a composite function of the above form, y'(x) can be found using the chain rule as follows:

In a composite function, the f(x) term is referred to as the outer function, while g(x) is referred to as the inner function. For example, let y(x) = sin(x^{3}). In this example, the outer function, f(x), is sin(x), and the inner function, g(x), is x^{3}. Given the above, y(x) can be differentiated using the chain rule as follows:

When using the chain rule, it is important to correctly identify the outer and inner functions. The outer function is always the operation that would be performed last if we were to evaluate the function. For example, given the function F(x) = sin^{2}(x), a common mistake would be to identify the outer function as sin(x) and the inner function as x^{2} when it is actually the opposite that is true. We can see this more clearly by rewriting sin^{2}(x) as [sin(x)]^{2}. The outer function is x^{2} since the exponent is the last operation we would compute in this case, making sin(x) the inner function.

Aside from correctly identifying the outer and inner functions of a composite function then applying the chain rule, the process of differentiation is otherwise very similar to differentiating individual functions. Below are some examples of using the chain rule with a few commonly encountered types of functions.

Examples

Differentiate the following functions using the chain rule.

Powers

i.

ii.

Trigonometric functions

i.

ii.

Composite functions made up of more than two functions can be differentiated by applying the chain rule multiple times. Let f, g, and h comprise a composite function y(x) such that y(x) = f(g(h(x))). The derivative, y'(x), can then be computed using the chain rule as follows:

Examples

Differentiate the following composite functions using the chain rule:

i.

ii.

## Implicit differentiation

The chain rule is used as part of implicit differentiation. Implicit differentiation involves differentiating equations with two variables by treating one of the variables as a function of the other. For example, given the equation

we can treat y as an implicit function of x and differentiate the equation as follows:

Note that the derivative of 3y^{5} with respect to x is 15y^{4} dy/dx, not just 15y^{4}. This is because y is treated as a function of x, calling for the use of the chain rule. Refer to the implicit differentiation page for a more complete explanation of implicit differentiation.