Chain rule

<math>\frac{d}{dx} \sin x=?\,</math>

This article/section deals with mathematical concepts appropriate for late high school or early college.

The chain rule in calculus is a formula for determining the derivative of a composite function:

<math>f(g(x))' = f'(g(x))\times g'(x)</math>

The chain rule can also be expressed as:

<math>\frac {dy}{dx} = \frac {dy} {du} \times \frac {du}{dx}.</math>

The chain rule can also be applied to multivariable functions. The derivative of a multivariable function is expressed as follows:

<math> \frac {d}{dt}(f(x(t), y(t))) = \frac{\partial f}{dx}\times \frac{dx}{dt} + \frac{\partial f}{dy}\times \frac{dy}{dt} </math>

or in vector notation:

<math> \nabla f \cdot \frac {dr}{dt} </math>

where r is the vector function

<math> r = \langle x(t), y(t), z(t) ... \rangle </math>

The function r is sometimes called the path of the particle.

Sometimes it might be helpful to change variables into a more convenient form before differentiating.