Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
More Math Help
Tutoring
Tools and Resources
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books

You CAN Ace Calculus

17calculus > derivatives > chain rule

 basic derivative rules power rule product rule quotient rule Some of the practice problems require you to know one or more of the following rules. If you haven't learned all these rules yet, no worries. You can filter out the problems that require techniques you do not know. trig derivatives exponential derivatives derivatives of logarithms

### Calculus Main Topics

Derivatives

Derivative Applications

Optimization

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

### Chain Rule FAQs

free ideas to save on books - bags - supplies 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.

Derivative - Chain Rule

The Chain Rule is probably the most important derivative rule that you will learn since you will need to use it a lot and it shows up in various forms in other derivatives and integration. It can also be a little confusing at first but if you stick with it, you will be able to understand it well.

There are two main ways to learn the chain rule, substitution and direct. I recommend writing out the substitution form while you are first learning. After you get the hang of it, you can use the direct (shorter) version.
NOTE: Do not be lazy here and try to use the direct version first. Write everything out so that you can see it. If you don't, you will struggle with this concept for the rest of your calculus life and this rule is used everywhere, and I mean EVERYWHERE from now on. You can't get away from it and you can't get around it. So, let's get started.

 What The Chain Rule Says

The chain rule says, if you have a function in the form y=f(u) where u is a function of x, then $$\displaystyle{ \frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx} }$$.
The notation tells you that $$u(x)$$ is a composite function of $$f$$.

 Substitution Method

This is the method you should learn first since it breaks the problem into smaller, bite-size pieces that you already know how to take derivatives of. The best way to learn this concept is with examples.

1. $$y=(x+1)^2$$ can be separated into $$u = x+1$$ and $$y=u^2$$. We started on the outside and noticed that there was something with a power of 2. So we took that something and set it equal to u.

2. $$y=\sin(x^2)$$ can be separated into $$u=x^2$$ and $$y=\sin(u)$$. We noticed that, on the outside, we have the sine function. So we set whatever was on the inside to u.

3. $$y=\sqrt{\sin(x)}$$ has an outside function as a square root, while the inside function is sine. So $$u=\sin(x)$$ and $$y = \sqrt{u}$$.

4. $$y=\sqrt{\sin(x^2)}$$ is a combination of the last two examples and is a little more complicated (but not much). We do the same thing as before and notice that the outside function a square root. So we set u to the inside function, i.e. $$u=\sin(x^2)$$ and $$y=\sqrt{u}$$. When we try to take the derivative $$du/dx$$, we need to do substitution again. So we let $$v=x^2$$ and $$u=\sin(v)$$.

In all of these examples, we just showed you how to set the problem with substitution. You would then complete the problem by taking the derivative of each piece. Try this example before going on to the direct method.

 Example - - Calculate the derivative of $$y=\cos(x^2+3)$$ using the substitution method.
 Direct Method

Okay, once you get the basic idea of the chain rule down, the next step is be able to write out the derivative without using substitution. Note: Some instructors jump to this technique right away without emphasizing the substitution concept. If your instructor does that, check your textbook and learn the chain rule with substitution first. You will be glad you did later on down the road.

Now, there are two ways to think about working problems without substitution. The technique that I think works best is to start from the outside and work your way in. Let's see how this is done using an example of an outside to inside technique.

 Example - - Calculate the derivative of $$g(t)=\sin(t^3)$$ using the direct method.

A second way to work this is to start on the inside and work your way out. Some of the videos further down on this page show this technique but we do not recommend it. The reason is that the more difficult problems (and ones you will probably see on your exam) are nested, i.e. you need to use the chain rule repeatedly. It is much easier to break the problem down and see what you need to do if you start on the outside and work your way in. Starting in the inside is sometimes difficult to find where to start. Let's do an example of this.

 Example - - Evaluate $$\displaystyle{ \frac{d}{dx}[\sin(\tan(2x))]}$$.

Before working practice problems, let's watch a video. This video has a great explanation of the chain rule at the first and then some examples that should help you understand how to use this rule.

 PatrickJMT - Chain Rule for Finding Derivatives

Okay, now you are ready for some practice problems.

### Search 17Calculus

practice filters

use basic derivatives only (9)

use trig rules (9)

use exponential and/or logarithmic rules (7)

Practice Problems

Instructions - - Unless otherwise instructed, calculate the derivative of these functions and give your final answers in completely factored form.

 Level A - Basic

Practice A01

$$\displaystyle{ 3e^{ x^2+7 } }$$

solution

Practice A02

$$\displaystyle{ (x^3+1)^4 }$$

solution

Practice A03

$$\displaystyle{ \ln(3x^2+9x-5) }$$

solution

Practice A04

Calculate the first three derivatives of $$f(x)=\ln(2+3x)$$.

solution

Practice A05

$$\displaystyle{ f(x)=\sqrt{2x+1} }$$

solution

Practice A06

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

solution

Practice A07

$$\displaystyle{ 7 \sin(x^2+1) }$$

solution

Practice A08

$$\displaystyle{ f(x) = (x^4+3x^2-2)^5 }$$

solution

Practice A09

$$\tan(x^5 + 2x^3 - 12x)$$

solution

Practice A10

$$y=4^{x^3-\sin x}$$

solution

Practice A11

$$\displaystyle{ g(x)= (1+4x)^5(3+x-x^2)^8 }$$

solution

Practice A12

$$\displaystyle{ f(x) = (2x-3)^4 (x^2+x+1)^5 }$$

solution

Practice A13

$$\displaystyle{ y=\sin^3(x) \tan(4x) }$$

solution

Practice A14

$$\displaystyle{ y = (2x^5-3) \sin(7x) }$$

solution

Practice A15

$$y=x^2\cos(1/x^3)$$

solution

Practice A16

$$(x^2-1)e^{-x}$$

solution

 Level B - Intermediate

Practice B01

$$\displaystyle{ y = (1+\cos^2(7x))^3 }$$

solution

Practice B02

$$\displaystyle{ y = \cos \left( \frac{1-e^{2x}}{1+e^{2x}} \right) }$$

solution

Practice B03

$$\displaystyle{ y=\left( \frac{x^2+1}{x^2-1} \right)^3 }$$

solution

Practice B04

$$\displaystyle{ g(t) = \frac{(t+4)^{1/2}}{(t-4)^{1/2}} }$$

solution

Practice B05

Use the quotient rule to find $$f'(x)$$ of $$\displaystyle{ f(x)=\frac{6}{\ln(8x^2)} }$$.

solution

Practice B06

$$\displaystyle{y=\ln\sqrt{\frac{x^2+1}{x+3}}}$$

solution

Practice C01

$$\displaystyle{ y=\left[ \tan \left( \sin \left( \sqrt{x^2+8x}\right) \right) \right]^5 }$$

solution

Practice C02

$$\displaystyle{ y=x \sin(1/x) + \sqrt[4]{(1-3x)^2 + x^5} }$$

solution

Practice C03

$$\displaystyle{ f(x)=\left[ \frac{x+4}{\sqrt{x^2+1}} \right]^3 }$$

solution