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# Power rule

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.1 (EK)

## 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 the limit as 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 fine 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 then 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 zero so n can be anything it can be positive and negative it could be it doesn't happen it does not have to be an integer the power rule tells us that the derivative of this the derivative of this F prime of X is just going to be equal to and so you're literally bringing this out front n times X 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 3rd 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 the mockingly straightforward so this is going to be three times X to the three minus one power or this is going to be equal to three X the 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 100 X to the negative 100 minus 1 which is equal to negative 100 X 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 two point five seven one power and we are we are concerned with what is Z prime of X well once again power rule simplifies our life and is two point five seven one so it's going to be two point five seven one times X to the two point five seven one minus one power so it's going to be equal to let me make sure I'm not falling off the bottom of the page two point five seven one times X to the one point five seven one power hopefully you enjoyed that 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
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