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Power rule

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.1 (EK)
Sal introduces the power rule, which tells us how to find the derivative of xⁿ. Created by Sal Khan.

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  • piceratops ultimate style avatar for user Everest Witman
    Does the power rule tell us how to deal with an expression with a coefficient, like f(x) = 2x^3? And what do I do if I have more than one term in my equation, like f(x) = 3x^2 + x + 3?
    (2 votes)
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    • Yes, you can use the power rule if there is a coefficient. In your example, 2x^3, you would just take down the 3, multiply it by the 2x^3, and make the degree of x one less. The derivative would be 6x^2.
      Also, you can use the power rule when you have more than one term. You just have to apply the rule to each term. In your example, f(x) = 3x^2 + x + 3, the derivative of f(x) would be 6x+1
      (149 votes)
  • leaf blue style avatar for user Sean
    What would the derivative be of something like 2^X? (two raised to the power of x) Would it still be 2x?
    (2 votes)
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    • mr pants teal style avatar for user SanFranGiants
      Polynomials are anything of the form x^n+x^n-1+... and you can have coefficients. But 2^x is exponential. Finding the derivative of an exponential function is a whole different process and you don't use the power rule. The derivative of e^x=e^x and the derivative of n^x where n is any number/coefficient is just n^x*ln(n)
      (4 votes)
  • leaf green style avatar for user Elijah Webb
    Does the power rule work If you have a function with a square root or a fraction?
    (2 votes)
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  • leaf green style avatar for user Simon Adams
    The test questions to this lesson include the following example:
    a= 2x^4 + 6x^3 - 7x^2
    and gives the answer (a prime) = -14x
    Why would the answer not also include the terms: 8x^3 + 18x^2?
    (5 votes)
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  • aqualine ultimate style avatar for user Stareomancer
    Why can't n = 0? If n is 0, then x^n is 1, right? Then its derivative is 0 like any other constant. And that follows the power rule doesn't it? So why can't n = 0?
    (2 votes)
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    • female robot grace style avatar for user Anna
      x^0 is 1 unless x = 0. Let me show you this with a couple of examples.

      Lets say that you have 100^0 what would that be? Well of course it would be 1.

      what about 0^0? That is the same as 0/0 which is undefined.

      so long story short x^n is possible as long as x is not equal to 0 when n = 0.
      (3 votes)
  • leaf blue style avatar for user Shar
    How do I figure out if a number can be expressed as a power greater than 1? Example 120, 400, 100, 250, 200 and how do you express the number as a power?
    (3 votes)
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    • male robot hal style avatar for user Sid
      You could factor each number and look for factors that occurs an even number of times.

      100 = 2 * 2 * 5 * 5 = (2 * 5) * (2 * 5) = 10 * 10 = 10^2
      400 = 2 * 2 * 2 * 2 * 5 * 5 = (2 * 2 * 5) * (2 * 2 * 5) = 20 * 20 = 20^2

      Some numbers will give you several options, e.g. 81 = 3^4 = 9^2.
      (3 votes)
  • marcimus pink style avatar for user Emma
    Is there ever a case where you take a derivative twice?
    So for example, x^3-5x^2+12.
    Using the power rule, you'd get 3x^2-10x
    Is there a case where you would apply the power rule again and get 6x-10?
    (2 votes)
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    • mr pants teal style avatar for user SanFranGiants
      Yes! In Physics, if f(x) represents the position of a particle (get familiar to all the problems involving particles if you're taking the AP Calc test/exam) then f'(x) represents the velocity of the particle (not to be confused with speed) and f''(x) or the second derivative of f(x) represents the acceleration of the particle. You use double derivatives A LOT in calc!
      (2 votes)
  • aqualine ultimate style avatar for user Raul Baez
    why isn't f'(x)= 1 / 100(x^99) when the f(x) is x^-100?
    (2 votes)
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    • piceratops tree style avatar for user Theresa Johnson
      The power rule is, for f(x) = x^n,
      f'(x) = n * x^(n-1).
      Now, if n is -100, then
      f'(x) = -100*x^(-100-1)
      first notice that -100-1 = -101 not -99, so
      f'(x) = -100*x^(-101)
      second, the negative power is on the x only, not the coefficient so it is the only thing that is flipped down to the denominator. The correct derivative is
      f'(x) = -100/x^101
      (3 votes)
  • blobby green style avatar for user Alex Kowalewski
    Isn't a correct way of writing the answer to d/dx(x^-100) also (-100)/(x^101)?
    (2 votes)
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  • male robot donald style avatar for user Michael Kupperman
    When asked for a proof of a derivative of a constant, can the power rule be used as where c represents the constant and x the variable:

    y=cx^0
    y'=c*0*x^-1
    Because there the entire term is multiplied by zero, the expression for the derivative is equal to 0?
    (2 votes)
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    • piceratops ultimate style avatar for user Just Keith
      I think that would be adequate, but you can more directly prove it from the definition of a derivative: since for a constant f(x) is the same value for all x , you have a 0 in the numerator. Thus, the derivative of all constants must be 0.
      (2 votes)

Video transcript

In this video, we will cover the power rule, which really simplifies our life when it comes to taking derivatives, especially derivatives of polynomials. You are probably already familiar with the definition of a derivative, limit is delta x approaches 0 of f of x plus delta x minus f of x, all of that over delta x. And it really just comes out of trying to find the slope of a tangent line at any given point. But we're going to see what the power rule is. It simplifies our life. We won't have to take these sometimes complicated limits. And we're not going to prove it in this video, but we'll hopefully get a sense of how to use it. And in future videos, we'll get a sense of why it makes sense and even prove it. So the power rule just tells us that if I have some function, f of x, and it's equal to some power of x, so x to the n power, where n does not equal 0. So n can be anything. It can be positive, a negative, it could be-- it does not have to be an integer. The power rule tells us that the derivative of this, f prime of x, is just going to be equal to n, so you're literally bringing this out front, n times x, and then you just decrement the power, times x to the n minus 1 power. So let's do a couple of examples just to make sure that that actually makes sense. So let's ask ourselves, well let's say that f of x was equal to x squared. Based on the power rule, what is f prime of x going to be equal to? Well, in this situation, our n is 2. So we bring the 2 out front. 2 times x to the 2 minus 1 power. So that's going to be 2 times x to the first power, which is just equal to 2x. That was pretty straightforward. Let's think about the situation where, let's say we have g of x is equal to x to the third power. What is g prime of x going to be in this scenario? Well, n is 3, so we just literally pattern match here. This is-- you're probably finding this shockingly straightforward. So this is going to be 3 times x to the 3 minus 1 power, or this is going to be equal to 3x squared. And we're done. In the next video we'll think about whether this actually makes sense. Let's do one more example, just to show it doesn't have to necessarily apply to only these kind of positive integers. We could have a scenario where maybe we have h of x. h of x is equal to x to the negative 100 power. The power rule tells us that h prime of x would be equal to what? Well n is negative 100, so it's negative 100x to the negative 100 minus 1, which is equal to negative 100x to the negative 101. Let's do one more. Let's say we had z of x. z of x is equal to x to the 2.571 power. And we are concerned with what is z prime of x? Well once again, power rule simplifies our life, n it's 2.571, so it's going to be 2.571 times x to the 2.571 minus 1 power. So it's going to be equal to-- let me make sure I'm not falling off the bottom of the page-- 2.571 times x to the 1.571 power. Hopefully, you enjoyed that. And in the next few videos, we will not only expose you to more properties of derivatives, we'll get a sense for why the power rule at least makes intuitive sense. And then also prove the power rule for a few cases.