The product rule is a rule for differentiating a product of two functions.

If \(y=uv\) is a product of two functions, then \[\frac{dy}{dx}=\dfrac{du}{dx}v+u\dfrac{dv}{dx}.\]

This can be generalised to a product of more than two functions. If \(y=uvw\ldots\), then \[\frac{dy}{dx}=\dfrac{du}{dx}vw\ldots+u\dfrac{dv}{dx}w\ldots+ uv\dfrac{dw}{dx}\ldots+\dotsb.\]

The Leibniz rule is a generalisation of the product rule to second and higher derivatives: \[\frac{d^n}{dx^n}(uv)=\sum_{k=0}^n {n \choose k} u^{(k)} v^{(n-k)},\] where \(u^{(k)}\) means the \(k\)th derivative of \(u\).