Product rule

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 $u$.