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 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.

WikiBooks - Quotient Rule

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

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 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
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17calculus > derivatives > quotient rule

 How To Use The Quotient Rule Things To Watch For Quotient Rule Proof [video] Practice

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   $$d(x) = x^2$$.

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

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

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

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

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

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

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

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

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

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

Problem Statement

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

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

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.

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

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

Problem Statement

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $$\displaystyle{\frac{7e^t}{5-9e^t}}$$.

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

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.

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

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

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

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

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

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

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

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

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