The bproduct rule/b 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 "ilike the product rule, except with a minus instead of a plus, all over the second function squared/i".

[b]uFunction of x in terms of y[/u]/b

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)} )

[b]uQuotient rule[/u]/b

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

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