Chain rule

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

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:

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

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.