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Power rule

Let's dive into the power rule, a handy tool for finding the derivative of xⁿ. This rule simplifies the process of taking derivatives, especially for polynomials, by bringing the exponent out front and decrementing the power. We explore examples with positive, negative, and fractional exponents. Created by Sal Khan.

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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.