Product and Quotient Rule

The Product Rule is a formula that we can use to differentiate the product of 2 (or more) functions. The Quotient Rule is for the quotient of two functions (one function divided by another). The rules are quite easy to apply. The challenge is often simplifying your answer so that you can find the coordinates of any stationary points. To do this you need to do some careful factorising. It is also important to be confident in using other differentiation techniques, like the Chain Rule. 


Key Concepts

On this page, you should learn about

  • the product rule for differentiation
  • the quotient rule for differentiation
Product Rule \(y=uv\\ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}\) \(f(x)=g(x)h(x)\\ f'(x)=g(x)h'(x)+h(x)g'(x)\)
     
Quotient Rule \(y=\frac { u }{ v } \\ \frac { dy }{ dx } =\frac { v\cdot \frac { du }{ dx } -u\cdot \frac { dv }{ dx } }{ v^{ 2 } } \) \(f(x)=\frac { g(x) }{ h(x) } \\ f'(x)=\frac { h(x)\cdot g'(x)-g(x)\cdot h'(x) }{ { \left[ h(x) \right] }^{ 2 } } \)

 

Essentials

The following videos will help you understand all the concepts from this page

Product Rule Easy

In the following example we look at using the product rule to find the derivative of a product of 2 simple functions


Given that \(y=x^2lnx\), find \(\frac{dy}{dx}\)

Notes from the video

Product Rule Hard

Applying the product rule is not difficult, but what can be challenging is simplifying the result so that stationary points can be found. In order to do this careful factorising is required. Here is an example that shows how to do it.


\(f(x)=x^3(x-3)^2\)

The function f has three stationary points. Find the x coordinates of these points


Notes from the video

Quotient Rule Easy

The Quotient Rule is fairly straight forward to apply. However, functions that require this rule often require good knowledge of other differentiation techniques, like the Chain Rule. The example below requires some algebraic simplifying techniques also.


Let \(f(x)=\frac{(3x-2)^2}{x^3} \quad,x\neq 0\)

Find \(f'(x)\)


Notes from the video

Quotient Rule Hard

One of the challenging things about the Quotient Rule is that we are often required to simplify our answers and leave in them in a factorised form so that we can find the coordinates of any stationary points. Here is a typical example where the simplifying requires careful algebraic manipulation.


The graph of \(y=\frac{2x}{\sqrt{x-1}} \quad ,x>1\) has a local minimum.

Find the coordinates of this point.


Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises The Product Rule


START QUIZ!

Here is a quiz that practises The Quotient Rule


START QUIZ!

Exam-style Questions

Question 1

Let \(y=xe^x\)

a) Find \(\frac{dy}{dx}\)

b) Show that \(\frac{d^2y}{dx^2}=e^x(2+x)\)

c) Find the coordinates of the stationary point and show that it is a local minimum.

Hint

Full Solution

 

Question 2

Let \(f(x)={ x }^{ 2 }{ (2x-3) }^{ 3 }\)

a) Find \(f'(x)\)

b) The graph of y = f(x) has stationary points at x = 0, x= \(\frac{3}{2}\) and x = a. Find the value of a

Hint

Full Solution

 

Question 3

Let \(f(x)=\frac{lnx}{x},x>0\)

a) Show that \(f'(x)=\frac{1-lnx}{x^2}\)

b) Find \(f''(x)\)

c) The graph of f has a point of inflexion at A. Find the x-coordinate of A.

Hint

Full Solution

 

Question 4

Let \(f(x)=tanx\). A\((\frac{\pi}{3},\sqrt{3})\) is a point that lies on the graph of \(y=f(x)\)

a) Given that \(tanx=\frac{sinx}{cosx}\) find \(f'(x)\)

b) Show that \(f'(\frac{\pi}{3})=4\)

c) Find the equation of the normal to the curve y = f(x) at the point A

d) Show that the normal crosses the y axis at \(\sqrt{3}+\frac{\pi}{12}\)

Hint

Full Solution

 

Question 5

Let \(f(x)=e^{2x}cosx\)

a) Find \(f'(x)\)

b) Show that \(f''(x)=4f'(x)-5f(x)\)

Hint

Full Solution

 

MY PROGRESS

How much of Product and Quotient Rule have you understood?