You CAN Ace Calculus

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 skip those problems and come back to them later.

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.

Related Topics and Links

external links you may find helpful

quotient rule youtube playlist

WikiBooks - Quotient Rule

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

free ideas to save on books

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.

How To Use The Quotient Rule

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 \( \displaystyle{f(x) = \frac{x+5}{x^2}} \).

\(\displaystyle{ f'(x) = \frac{-(x+10)}{x^3}}\)

Problem Statement - Calculate the derivative of \( \displaystyle{f(x) = \frac{x+5}{x^2}} \).
Solution - In this case \( n(x) = x+5 \)   and   \( d(x) = x^2 \).

\(\displaystyle{ f'(x) = \frac{(x^2)(1) - (x+5)(2x)}{(x^2)^2} = }\) \(\displaystyle{ \frac{x[x-2(x+5)]}{x^4} = }\) \(\displaystyle{ \frac{x-2x-10}{x^3} = }\) \(\displaystyle{ \frac{-(x+10)}{x^3}}\)

Final Answer

\(\displaystyle{ f'(x) = \frac{-(x+10)}{x^3}}\)

close solution

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.

  1. 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.

  2. 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.

  3. 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 [6min-27secs]

video by MIP4U

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

 
chain rule →

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Derivative Quotient Rule - Practice Problems Conversion

[A01-941] - [A02-942] - [A03-943] - [A04-944] - [A05-945] - [A06-946] - [A07-947] - [A08-948] - [A09-949]

[A10-950] - [A11-951] - [A12-952] - [A13-954] - [A14-955] - [A15-1313] - [B01-953]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

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

Here are some problems that use only the quotient rule and the basic rules discussed on the main derivatives page (power rule, constant rule and constant multiple rule).

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{x+1}{x-1}}\).

Solution

949 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{x^2-4}{x^2+4}}\).

Solution

950 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{3}{x^2+x+1}}\).

Solution

951 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{1}{x+1}-\frac{1}{x-1}}\).

Solution

948 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{x^3-5x^2+7x+3}{x^2+9}}\).

Final Answer

\( \displaystyle{ \frac{d}{dx} \left[ \frac{x^3-5x^2+7x+3}{x^2+9} \right] = \frac{x^4+20x^2-96x+63}{(x^2+9)^2} } \)

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{x^3-5x^2+7x+3}{x^2+9}}\).

Solution

\(\displaystyle{\frac{d}{dx}\left[\frac{x^3-5x^2+7x+3}{x^2+9}\right] }\)

Apply the quotient rule.

\(\displaystyle{\frac{(x^2+9)d[x^3-5x^2+7x+3]/dx - (x^3-5x^2+7x+3)d[x^2+9]/dx}{(x^2+9)^2} }\)

Take the derivatives.

\(\displaystyle{\frac{(x^2+9)(3x^2-10x+7)-(x^3-5x^2+7x+3)(2x)}{(x^2+9)^2} }\)

Look for common terms in the numerator. In this case, there are not any, so just multiply out.

\(\displaystyle{\frac{(3x^4 -10x^3+7x^2+27x^2-90x+63)-(2x^4-10x^3+14x^2+6x)}{(x^2+9)^2} }\)

\(\displaystyle{\frac{x^4+20x^2-96x+63}{(x^2+9)^2}}\)

After completing all derivatives, we always check for common terms between the two main factors in the numerator to factor out. In this case, there were none, so our only option was to multiply out. Notice we leave the denominator in its most compact and factored form.

Final Answer

\( \displaystyle{ \frac{d}{dx} \left[ \frac{x^3-5x^2+7x+3}{x^2+9} \right] = \frac{x^4+20x^2-96x+63}{(x^2+9)^2} } \)

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{x^2-3x+7}{\sqrt{x}}}\).

Final Answer

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{x^2-3x+7}{\sqrt{x}}}\).

Solution

\(\begin{array}{rcl} & & \frac{d}{dx} \left[ \frac{x^2-3x+7}{\sqrt{x}} \right ] \\ & = & \frac{x^{1/2}(2x-3) - (x^2-3x+7)(1/2)x^{-1/2}}{x} \\ & = & \frac{x(2x-3) - (1/2)(x^2-3x+7)}{x^{3/2}} \\ & = & \frac{2x(2x-3)-(x^2-3x+7)}{2x^{3/2}} \\ & = & \frac{3x^2-3x-7}{2x^{3/2}} \end{array}\)

In the first line, we used the quotient rule. After that, just algebra. The tricky algebra here involves \(x^{-1/2}\). In the second line, we have x's in both of the main terms in the numerator. We need to factor those out. So we multiply the numerator and the denominator by \( x^{1/2} \). We chose \( x^{1/2} \) since our goal is to have all positive powers in our answer and multiplying \( x^{-1/2} \) by \( x^{1/2} \) gives us \( x^{-1/2} \cdot x^{1/2} = x^{-1/2+1/2} = x^0 = 1 \). This gets rid of the \( x^{-1/2} \) in the second term.

Final Answer

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

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{y=\frac{t^2+2}{t^4-3t^2+1}}\).

Solution

946 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{x-1}{x^2+2x+1}}\).

Solution

947 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{y=\frac{45}{5+x+\sqrt{x}}}\).

Solution

952 solution video

video by Krista King Math

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{y=\frac{x^2+1}{x^5+x}}\) .

Solution

954 solution video

video by PatrickJMT

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{x^2+3x}{x+4}}\).

Solution

1313 solution video

video by PatrickJMT

close solution

Here are a couple of problems that are a little more difficult but, after working the above problems successfully, you should have no problem with these.

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{x}{x+c/x}}\).

Solution

953 solution video

video by Krista King Math

close solution

\(\displaystyle{ h(x)=\frac{2x^3~k(x)}{3x+2} }\)

Problem Statement

\(\displaystyle{ h(x)=\frac{2x^3~k(x)}{3x+2} }\)

Hint

The function \(k(x)\) is not known. So when you take the derivative just write \(k'(x)\).

Problem Statement

\(\displaystyle{ h(x)=\frac{2x^3~k(x)}{3x+2} }\)

Final Answer

\(\displaystyle{ h'(x) = \frac{2x^2[ 6k(x)(x+1) + xk'(x)(3x+2) ]}{(3x+2)^2} }\)

Problem Statement

\(\displaystyle{ h(x)=\frac{2x^3~k(x)}{3x+2} }\)

Hint

The function \(k(x)\) is not known. So when you take the derivative just write \(k'(x)\).

Solution

2225 solution video

video by MathTV

Final Answer

\(\displaystyle{ h'(x) = \frac{2x^2[ 6k(x)(x+1) + xk'(x)(3x+2) ]}{(3x+2)^2} }\)

close solution

These problems require you to know how to take the derivative of exponential functions.

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{7e^t}{5-9e^t}}\).

Final Answer

\( \displaystyle{ \frac{35e^t}{(5-9e^t)^2} }\)

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{7e^t}{5-9e^t}}\).

Solution

\( \begin{array}{rcl} & & \frac{d}{dt} \left[ \frac{7e^t}{5-9e^t} \right] \\ & = & \frac{(5-9e^t)d[7e^t]/dt - (7e^t)d[5-9e^t]/dt}{(5-9e^t)^2} \\ & = & \frac{(5-9e^t)(7e^t) - 7e^t(-9e^t)}{(5-9e^t)^2} \\ & = & \frac{(7e^t)(5-9e^t +9e^t)}{(5-9e^t)^2} \\ & = & \frac{35e^t}{(5-9e^t)^2} \end{array} \)

In the first step, we used the quotient rule. Following that, we used algebra to factor and simplify. Notice that between lines 3 and 4, we factored out \( 7e^t \) before multiplying out the numerator.

Final Answer

\( \displaystyle{ \frac{35e^t}{(5-9e^t)^2} }\)

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{f(x)=\frac{7x^2+1}{e^x}}\).

Solution

945 solution video

video by Krista King Math

close solution

These problems require you to know how to take the derivative of trig functions.

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{\sin(x)}{1+\cos(x)}}\).

Final Answer

\(\displaystyle{ \frac{d}{dx}\left[ \frac{\sin(x)}{1+\cos(x)} \right] = \frac{1}{1+\cos(x)}}\)

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{\frac{\sin(x)}{1+\cos(x)}}\).

Solution

\(\displaystyle{ \frac{d}{dx}\left[ \frac{\sin(x)}{1+\cos(x)} \right] }\)

\(\displaystyle{ \frac{[1+\cos(x)]d[\sin(x)]/dx - [\sin(x)]d[1+\cos(x)]/dx}{[1+\cos(x)]^2} }\)

\(\displaystyle{ \frac{[1+\cos(x)][\cos(x)] - [\sin(x)][-\sin(x)]}{[1+\cos(x)]^2} }\)

\(\displaystyle{ \frac{\cos(x)+\cos^2(x)+\sin^2(x)}{[1+\cos(x)]^2} }\)

Use the identity \( \sin^2(x) + \cos^2(x) = 1 \).

\(\displaystyle{ \frac{1+\cos(x)}{[1+\cos(x)]^2} }\)

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

Final Answer

\(\displaystyle{ \frac{d}{dx}\left[ \frac{\sin(x)}{1+\cos(x)} \right] = \frac{1}{1+\cos(x)}}\)

close solution

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

Problem Statement

Use the quotient rule to calculate the derivative of \(\displaystyle{y=\frac{\tan(x)}{x^{3/2}+5x}}\).

Solution

955 solution video

video by PatrickJMT

close solution
Real Time Web Analytics