Chain rule

$\frac{d}{dx} \sin x=?\,$

This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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

$f(g(x))' = f'(g(x))\times g'(x)$

The chain rule can also be expressed as:

$\frac {dy}{dx} = \frac {dy} {du} \times \frac {du}{dx}.$

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

$\frac {d}{dt}(f(x(t), y(t))) = \frac{\partial f}{dx}\times \frac{dx}{dt} + \frac{\partial f}{dy}\times \frac{dy}{dt}$

or in vector notation:

$\nabla f \cdot \frac {dr}{dt}$

where r is the vector function

$r = \langle x(t), y(t), z(t) ... \rangle$

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.