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17calculus > derivatives > quotient rule

Topics You Need To Understand For This Page

basic derivative rules

power rule

product rule

Some of the practice problems require you to know the following rules also (in their basic form, not including the chain rule). If you don't know one or more of these rules, no worries. You can filter them from the list of practice problems.

exponential derivative

derivatives of trig functions

You do NOT need to know the chain rule for anything on this page, including practice problems. For practice problems using the quotient rule and chain rule, see the chain rule page.

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ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com.

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Quotient Rule

When you are first learning the quotient rule, it is a good idea to write out intermediate steps. This rule is easy to get confused about, so writing out intermediate steps will help you get your head around the details. Later, you will be able to do more in your head and less on paper.

The quotient rule looks like this. If you have a function \(\displaystyle{f(x) = \frac{n(x)}{d(x)}} \), the quotient rule says the derivative is \(\displaystyle{ f'(x) = \frac{d \cdot n' - n \cdot d'}{d^2} }\). In this equation, we have used \(n(x)\) to denote the expression in the numerator and \(d(x)\) to denote the expression in the denominator.

Okay, so this seems pretty straight-forward. Let's do an example. Try this on your own before looking at the solution.

Calculate the derivative of \( \displaystyle{f(x) = \frac{x+5}{x^2}} \).

A Note About Simplifying

Things To Watch For

There are a few things we see students do quite often that we want to warn you about.
- First, go back to the example and notice that we take the denominator and assign it directly to d(x). I have seen students think \( d(x) = x^{-2} \). This is NOT the case. We do not take the denominator, move it to the numerator and call that d(x). So be very careful here.
- Second, again, go back to the example and notice that, when simplifying, the first thing we do in the numerator is look for a common factor between the two terms. In the example, we had a factor of x. We factored it out and canceled it with an x in the denominator. This happens quite often when using the quotient rule.
- Third, the quotient rule itself is not that difficult to do. The thing that will probably trip you up the most is the algebra you have to do with simplifying. So this technique will often challenge you to remember and use your algebra rules related to factoring and powers.

Before working some practice problems, take a few minutes and watch this video showing a proof of the quotient rule.

MIP4U - Proof of the Quotient Rule

Okay, it is time for some practice problems. After that, the chain rule is next.

 
chain rule →

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use basic derivatives and quotient rule only

use trig rules

use exponential and/or logarithmic rules

Practice Problems

Instructions - - Unless otherwise instructed, calculate the derivatives of the following functions using the quotient rule, giving your answers in simplified form.

Level A - Basic

Practice A01

\(\displaystyle{\frac{x^3-5x^2+7x+3}{x^2+9}}\)

answer

solution

Practice A02

\(\displaystyle{\frac{x^2-3x+7}{\sqrt{x}}}\)

answer

solution

Practice A03

\(\displaystyle{\frac{7e^t}{5-9e^t}}\)

answer

solution

Practice A04

\(\displaystyle{\frac{\sin(x)}{1+\cos(x)}}\)

answer

solution

Practice A05

\(\displaystyle{f(x)=\frac{7x^2+1}{e^x}}\)

solution

Practice A06

\(\displaystyle{y=\frac{t^2+2}{t^4-3t^2+1}}\)

solution

Practice A07

\(\displaystyle{f(x)=\frac{x-1}{x^2+2x+1}}\)

solution

Practice A08

\(\displaystyle{f(x)=\frac{1}{x+1}-\frac{1}{x-1}}\)

solution

Practice A09

\(\displaystyle{f(x)=\frac{x+1}{x-1}}\)

solution

Practice A10

\(\displaystyle{f(x)=\frac{x^2-4}{x^2+4}}\)

solution

Practice A11

\(\displaystyle{f(x)=\frac{3}{x^2+x+1}}\)

solution

Practice A12

\(\displaystyle{y=\frac{45}{5+x+\sqrt{x}}}\)

solution

Practice A13

\(\displaystyle{y=\frac{x^2+1}{x^5+x}}\)

solution

Practice A14

\(\displaystyle{y=\frac{\tan(x)}{x^{3/2}+5x}}\)

solution

Practice A15

\(\displaystyle{f(x)=\frac{x^2+3x}{x+4}}\)

solution


Level B - Intermediate

Practice B01

\(\displaystyle{f(x)=\frac{x}{x+c/x}}\)

solution

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