Differentiation of a Product, Quotient and Simple Cases of a Function of a Function

Differentiation of a Product

The Product Rule states that the derivative of the product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first. More formally, if f(x) = u(x)v(x), then f’(x) = u(x)v’(x) + v(x)u’(x).
This rule can be easily extended to the product of more than two functions.
When differentiating products, it’s important to remember to apply the product rule correctly. Check your work by re-differentiating your answer.

Differentiation of a Quotient

The Quotient Rule is used to differentiate the quotient of two functions. If f(x) = u(x)/v(x), then f’(x) = [v(x)u’(x) - u(x)v’(x)] / [v(x)]^2.
Be careful to correctly apply the quotient rule, especially being sure to square the denominator function, v(x), in the formula.

Differentiation of a Function of a Function

Sometimes functions are composed of more than one function, a situation often referred to as a function of a function. The chain rule can be used to differentiate these composite functions.
The Chain Rule states that if f(x) = g(h(x)), then f’(x) = g’(h(x)) * h’(x).
The outer function g is differentiated normally, but the inner function h(x) is left as it is. This derivative is then multiplied by the derivative of the inner function h(x).
The chain rule is usually applied multiple times when differentiating complex functions.

General tips

Practice applying the Product, Quotient, and Chain rules individually and in combination to ensure you become adept at identifying when each one is required.
Keep all the basic differentiation rules to hand - product, quotient and chain rules.
Remember to check your answers by re-differentiating if time allows, as this is the best way to ensure you have applied the rules correctly.
Always reduce your final answer to the simplest form possible for full marks.

The process might seem complex at first, but keep practising and it’ll become second nature. You can then apply these rules to undertake any differentiation task for your Calculus studies.