The product rule is used for differentiating functions divided by each other.

\(y = \dfrac{f(x)}{g(x)} \)

\(\boxed{\dfrac{dy}{dx} = \dfrac{f'(x)g(x) - f(x)g'(x)}{\big(g(x)\big)^2}} \)

Using substitution, if \(u = f(x) \) and \(v = g(x) \), then:

\(\boxed{\dfrac{dy}{dx} = \dfrac{\dfrac{du}{dx}v - u\dfrac{dv}{dx}} {v^2}} \)

Tip: Think of it as "like the product rule, except with a minus instead of a plus, all over the second function squared".

Function of x in terms of y

For functions of the form \(x = f(y) \), \(\dfrac{dy}{dx}\) can be found by differentiating with respect to \(y\) and finding the reciprocal of the result.

\(x = f(y) \)

\(\dfrac{dy}{dx} = \dfrac{1}{\Big(\dfrac{dx}{dy}\Big)} \)