Product, quotient and chain rules

The idea

Chain rule: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$. Product rule: $(uv)' = u'v + uv'$. Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$.

1. Spot the structure: a function inside a function (chain), two multiplied (product) or divided (quotient).
2. Write $u$, $v$ and their derivatives.
3. Substitute into the rule and simplify.

Watch out In the chain rule, multiply by the derivative of the inside.