What is a situation where you would need to use the constant multiple rule? Aren't all of the problems where you could use the constant multiple rule covered by the power rule? For instance: If you wanted to differentiate 3x^2, you could simply multiply 2*3 and then subtract 1 from the exponent. However, Sal uses the constant multiple rule in many of his earlier examples to first pull out the 3 and then find the derivative of x^2. It seems like the constant multiple rule is an unnecessary step when finding derivatives.

The power rule essentially tells you how to differentiate x^n. It doesn't say anything about multiples of x^n, like 3x^2. So when you differentiate 3x^2, you really apply both the power rule and the constant multiple rule. First, the constant multiple rule tells us that the derivative of 3x^2 is 3 times the derivative of x^2. Then, the power rule tells us that the derivative of x^2 is 2x. Put those two together, and you get that the derivative of 3x^2 is 6x. Once you're experienced in differentiating various functions, you can "forget" the particular rules you use, and just do it fluently. So you quickly differentiate 3x^2 by multiplying 2 by 3 and reducing the 2nd power to 1. This doesn't mean you're not using the underlying rules, it just means you don't need to acknowledge them anymore.

I know that pi is a constant, then f(x) = pi*x^2 has the differentiation f'(x) = 2*pi*x. But what about 'e' (Euler)? its something pi? Its a constant? f(x) = e*x^2 with f'(x) = 2*e*x. And, for what I use 'e'? And his differentiation: prime 'e'? Thanks a lot!

e is a real number, just like π, √2, or 7. It's just a constant. All the derivatives you gave are correct.

Main content

AP®︎ Calculus AB (2017 edition)

Course: AP®︎ Calculus AB (2017 edition) > Unit 3

Lesson 4: Derivatives of negative and fractional powers with power rule

Power rule review

Google Classroom

Review your knowledge of the Power rule for derivatives and solve problems with it.

What is the Power rule?

The Power rule tells us how to differentiate expressions of the form x, start superscript, n, end superscript (in other words, expressions with x raised to any power):

start fraction, d, divided by, d, x, end fraction, x, start superscript, n, end superscript, equals, n, dot, x, start superscript, n, minus, 1, end superscript

Basically, you take the power and multiply it by the expression, then you reduce the power by 1.

Want to learn more about the Power rule? Check out this video.

Differentiating polynomials

The Power rule, along with the more basic differentiation rules, allows us to differentiate any polynomial. Consider, for example, the monomial 3, x, start superscript, 7, end superscript. We can differentiate it as follows:

\begin{aligned} \dfrac{d}{dx}[3x^7]&=3\dfrac{d}{dx}(x^7)\quad\gray{\text{Constant multiple rule}} \\\\ &=3(7x^6)\quad\gray{\text{Power rule}} \\\\ &=21x^6 \end{aligned}

Problem 1

Current

f, left parenthesis, x, right parenthesis, equals, x, start superscript, 5, end superscript, plus, 2, x, cubed, minus, x, squared

f, prime, left parenthesis, x, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

Differentiating negative powers

The Power rule also allows us to differentiate expressions like start fraction, 1, divided by, x, squared, end fraction, which is basically x raised to a negative power. Consider this differentiation of start fraction, 1, divided by, x, squared, end fraction:

\begin{aligned} \dfrac{d}{dx}\left(\dfrac{1}{x^2}\right)&=\dfrac{d}{dx}(x^{-2})\quad\gray{\text{Rewrite as power}} \\\\ &=-2\cdot x^{-3}\quad\gray{\text{Power rule}} \\\\ &=-\dfrac{2}{x^3}\quad\gray{\text{Rewrite as fraction}} \end{aligned}

Problem 1

Current

start fraction, d, divided by, d, x, end fraction, left parenthesis, start fraction, minus, 2, divided by, x, start superscript, 4, end superscript, end fraction, plus, start fraction, 1, divided by, x, cubed, end fraction, minus, x, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

Differentiating fractional powers and radicals

The Power rule also allows us to differentiate expressions like square root of, x, end square root or x, start superscript, start superscript, start fraction, 2, divided by, 3, end fraction, end superscript, end superscript. Consider this differentiation of square root of, x, end square root:

\begin{aligned} \dfrac{d}{dx}\sqrt x&=\dfrac{d}{dx}\left(x^{^{\Large\frac{1}{2}}}\right)\quad\gray{\text{Rewrite as power}} \\\\ &=\dfrac{1}{2}\cdot x^{^{\Large-\frac{1}{2}}}\quad\gray{\text{Power rule}} \\\\ &=\dfrac{1}{2\sqrt x}\quad\gray{\text{Rewrite as radical}} \end{aligned}

Problem 1

Current

f, left parenthesis, x, right parenthesis, equals, 6, x, start superscript, start superscript, start fraction, 2, divided by, 3, end fraction, end superscript, end superscript

f, prime, left parenthesis, x, right parenthesis, equals

Want to try more problems like this? Check out these exercises:

Fractional powers differentiation
Radical functions differentiation intro

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Tanner Thygerson
Posted 7 years ago. Direct link to Tanner Thygerson's post “What is a situation where...”
What is a situation where you would need to use the constant multiple rule? Aren't all of the problems where you could use the constant multiple rule covered by the power rule?

For instance: If you wanted to differentiate 3x^2, you could simply multiply 2*3 and then subtract 1 from the exponent. However, Sal uses the constant multiple rule in many of his earlier examples to first pull out the 3 and then find the derivative of x^2. It seems like the constant multiple rule is an unnecessary step when finding derivatives.
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(5 votes)
Answer
- Tomer Gal
  Posted 7 years ago. Direct link to Tomer Gal's post “The power rule essentiall...”
  The power rule essentially tells you how to differentiate x^n. It doesn't say anything about multiples of x^n, like 3x^2. So when you differentiate 3x^2, you really apply both the power rule and the constant multiple rule.
  
  First, the constant multiple rule tells us that the derivative of 3x^2 is 3 times the derivative of x^2. Then, the power rule tells us that the derivative of x^2 is 2x. Put those two together, and you get that the derivative of 3x^2 is 6x.
  
  Once you're experienced in differentiating various functions, you can "forget" the particular rules you use, and just do it fluently. So you quickly differentiate 3x^2 by multiplying 2 by 3 and reducing the 2nd power to 1. This doesn't mean you're not using the underlying rules, it just means you don't need to acknowledge them anymore.
  Button navigates to signup page
  (37 votes)
Pedro IPF
Posted 7 years ago. Direct link to Pedro IPF's post “I know that pi is a const...”
I know that pi is a constant, then f(x) = pi*x^2 has the differentiation f'(x) = 2*pi*x. But what about 'e' (Euler)? its something pi? Its a constant? f(x) = e*x^2 with f'(x) = 2*e*x.
And, for what I use 'e'? And his differentiation: prime 'e'?
Thanks a lot!
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(2 votes)
Answer
- kubleeka
  Posted 7 years ago. Direct link to kubleeka's post “e is a real number, just ...”
  e is a real number, just like π, √2, or 7. It's just a constant. All the derivatives you gave are correct.
  Button navigates to signup page
  (4 votes)
Ayesha Malik
Posted 2 years ago. Direct link to Ayesha Malik's post “(x^2 + 1)^2 how can i sol...”
(x^2 + 1)^2 how can i solve this using the power rule im stuck :((
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(2 votes)
Answer
- Tabby
  Posted 2 years ago. Direct link to Tabby's post “To solve (x^2+1)^2, You h...”
  To solve (x^2+1)^2, You have to multiply the power rule equation by its derivate.
  For example, the ^2 on the outside will then move to the front of the function as part of the power rule.
  
  So, 2(x^2+1) * D/DX (x^2+1). After that, you can find the derivate for each separate part of the function. So, d/dx of (x^2)=2x and d/dx of (1)=0. Finally, your answer would be 2(x^2+1)*(2x), where 2 is distributed to the 2x.
  
  Thus your answer is f'(x)= 4x(x^2+1) which can be further reduced to 4x^3+4x.
  
  Hope this helps ^u^
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  (1 vote)
21eppleg
Posted 3 years ago. Direct link to 21eppleg's post “Can you use the power rul...”
Can you use the power rule for terms like (5x + 1)? If this were, say, raised to the 4th power would you be able to use the power rule, or would you have to use the chain rule?
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(2 votes)
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cing.lun.3
Posted 7 days ago. Direct link to cing.lun.3's post “Can you please help me so...”
Can you please help me solve F(t)=(5t-8)^2?
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(1 vote)
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Lumbini
Posted 2 years ago. Direct link to Lumbini's post “Does it apply to constant...”
Does it apply to constants? Because both constant and the term x have the same power but different derivatives
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(1 vote)
Answer
Janak
Posted 6 years ago. Direct link to Janak's post “How was the power rule de...”
How was the power rule derived?
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(2 votes)
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- Smith
  Posted 6 years ago. Direct link to Smith's post “https://www.khanacademy.o...”
  https://www.khanacademy.org/math/differential-calculus/basic-differentiation-dc/power-rule-dc/v/proof-d-dx-x-n
  https://www.khanacademy.org/math/differential-calculus/basic-differentiation-dc/power-rule-dc/v/proof-d-dx-sqrt-x
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  (0 votes)
Jorge Jiménez
Posted a year ago. Direct link to Jorge Jiménez's post “Can the exponent n be zer...”
Can the exponent n be zero? The function x^0 is 1 for all x, except maybe when x=0. Its derivative is 0 because is a constant. And the power rule says its derivative is 0•x^(0-1) which is 0 for all x values, but it's not defined when x=0.
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(1 vote)
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1tusharthakur1
Posted 4 years ago. Direct link to 1tusharthakur1's post “Why can't I use power rul...”
Why can't I use power rule, when a constant is raised to power?
For Example: a^x
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(1 vote)
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hashtagx
Posted 5 years ago. Direct link to hashtagx's post “When using the power rule...”
When using the power rule on a polynomial to a negative power, do you take the derivative of each of the terms on the inside? I've been able to take the derivative of the exponent on the outside, but I don't know what to do from there. For example, would the derivative of 1/(x-1) be -x^-2 or -(x-1)^-2?
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