17calculus
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
Radius of Convergence
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
Gradients
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
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > derivatives > power rule

Topics You Need To Understand For This Page

Calculus Main Topics

Tools

Related Topics and Links

Power Rule

on this page: ► how to use the power rule     ► when you can't use the power rule     ► higher order derivatives

Power Rule Theorem

For any rational number r, the function \(g(x)=x^r\) is differentiable, and \(\displaystyle{ \frac{d}{dx}[g(x)] = rx^{r-1} }\).

[ Note: We use the letter r here to emphasize that the exponent must be a rational number, i.e. it cannot be irrational, like \(\pi\), or a variable or anything other than a rational number. This detail will be extremely important very soon in your study of derivatives. ]

Search 17Calculus

How To Use The Power Rule

Basically the Power Rule Theorem says, if you have a term of the form \(x^r\) where r is a rational number, then the derivative of this is \(rx^{r-1}\). You think of this as taking the exponent and putting it in front of the term and then subtracting one from the exponent like this:
power rule
It helps to remember if you visualize the exponent moving, which is what we showed with the arrow. This idea works fine as long as there is no number in front of the x. If there is a number in front of the x, you just multiply the two numbers using the constant multiple rule. It looks like this: \( \displaystyle{ \frac{d}{dx}[ax^n] = a(nx^{n-1}) }\).
The reason it works like this is because we use the constant multiple rule first, then the power rule. It looks like this.

\(\displaystyle{ \frac{d}{dx}[ax^n] }\)

\(=\)

\(\displaystyle{ a\frac{d}{dx}[x^n] }\)

Constant Multiple Rule

\(=\)

\(\displaystyle{ a(nx^{n-1}) }\)

Power Rule

Constant Multiple Rule

\(\displaystyle{ \frac{d}{dx}[ax^n] = }\) \(\displaystyle{ a\frac{d}{dx}[x^n] }\)

Power Rule

\(\displaystyle{ a\frac{d}{dx}[x^n] }\) \(\displaystyle{ a(nx^{n-1}) }\)

Okay, let's do some examples.

Example 1:

\(\displaystyle{\frac{d}{dx}[x^3] = }\) \(\displaystyle{ 3x^{3-1} = 3x^2}\)

Example 2:

\(\displaystyle{\frac{d}{dx}[2x^3] = }\) \(\displaystyle{ 2\frac{d}{dx}[x^3] = }\) \(\displaystyle{ 2(3x^2) = 6x^2}\)

As you get more practice, you probably won't need to do the two middle steps as shown in Example 2. You will probably just think of taking the coefficient 2 times the exponent 3 to get 6 out front.

When You Can't Use The Power Rule

There are two main situations where you can not use the power rule when you have an exponent.
First, when the term that has a power does not exactly match the derivative variable. In this case, the chain rule is required. Here are two examples.

Example 3: \(\displaystyle{\frac{d}{dx}[y^3]}\)
Notice here that the variable 'y' does not match the 'x' in 'dx'. So you can't use the power rule here. You need to use the chain rule.

Example 4: \(\displaystyle{\frac{d}{dx}\left[ (x^2+5)^3\right]}\)
In this case, the term \( (x^2+5) \) does not exactly match the x in dx. So you can't use the power rule here either ( on the \(3\) power ). The chain rule is required.

The second main situation is when the exponent is not a rational number.
This is an important detail to remember (and an example of why you need to read theorems carefully). For example, you can't use the power rule on \( g(t) = 2^t \) because t is not a rational number. Notice the difference between \( 2^t \) [cannot use power rule] and \( t^2 \) [can use power rule].
Also, you cannot use the power rule on \( x^{\pi} \) since \( \pi \) is not rational.
In these cases logarithmic differentiation is the way to go.

Okay, before working practice problems, take a few minutes to just sit back and watch this video. It shows lots of examples using the power rule and the basic rules (constant rule, constant multiple rule and addition and subtraction rules).

PatrickJMT - Basic Derivative Examples

Now that you know the power rule and the basic rules on the main derivatives page, you will not use the limit definition to calculate derivatives, unless otherwise instructed to do so.

Higher Order Derivatives

Although higher order derivatives are not directly related to the power rule, you need to know the power rule in order to be able to understand examples of how to find higher order derivatives. So we include the discussion here.

Higher order derivatives are not very difficult. The idea is that, after taking the first derivative, you can take the derivative again to get the second derivative, and so on. Here is a quick example.
Example - - Find the second derivative of \(f(x)=3x^5 + 2x+1\).
The first derivative is \( f'(x) = 15x^4 + 2\).
Taking the derivative again, gives us the second derivative \( f''(x) = 60x^3 \).

Notation - - We can continue taking as many derivatives as we want. The third derivative is written \( f'''(x) \) or \( f^{(3)}(x)\). Notice that the derivative number is written in parentheses so that we can tell we are talking about a derivative and not a power, which we would write \( f^{3}(x)\). Starting with the fourth derivative, we abandon the 'multiple quote' notation and write \(f^{(4)}(x) \) and so on.

Okay, your next task is to work some practice problems.
After that, if you need a suggestion on where to go next, the product rule is the logical next step.

 
product rule →

Practice Problems

Instructions - - Unless otherwise instructed, calculate the derivatives of the following functions using the power rule and the basic rules on the derivatives page (but not the limit definition of the derivative) giving your answers in simplified form.

Level A - Basic

Practice A01

calculate \(f''(x)\) of \(f(x)=3x^4+3x^2+2x \)

answer

solution

Practice A02

\(\displaystyle{y=2x^2+3x-17}\)

solution

Practice A03

\(\displaystyle{f(x)=100-16x^2}\)

solution

Practice A04

\(\displaystyle{f(x)=4x-5}\)

solution

Practice A05

\(\displaystyle{y=x^6-7x^4+5/x}\)

solution

Practice A06

\(\displaystyle{y=3x^2}\)

answer

solution

Practice A07

\(\displaystyle{y=7x^3}\)

answer

solution

Practice A08

\(\displaystyle{y=5x^4}\)

answer

solution

Practice A09

\(\displaystyle{g(y)=y^4}\)

answer

solution

Practice A10

\(\displaystyle{f(x)=x^{10}}\)

solution

Practice A11

\(\displaystyle{g(x)=x^{5/2}+3x+4}\)

solution


Level B - Intermediate

Practice B01

\(\displaystyle{h(x)=\frac{1}{\sqrt[3]{x}}}\)

solution

Practice B02

\(\displaystyle{p(q)=\frac{-3}{\pi}\sqrt{q^{4.1}}}\)

solution

Practice B03

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

solution

Practice B04

What is \(f'(64)\) for \(\displaystyle{f(x)=x+3\sqrt{x}+4\sqrt[3]{x}}\)?

solution

Practice B05

\(\displaystyle{y=\sqrt[5]{x}+4\sqrt{x^5}}\)

solution

Practice B06

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

solution

Real Time Web Analytics
menu top search practice problems
17
menu top search practice problems 17