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Welcome today. We are discussing the power rule of derivatives, which in order to basically understand the material in this lesson, it is advisable that you first be familiar with the limit rule of limit definition of derivatives, the constant multiple rule of derivatives and the sum rule of derivatives. So the lastly are quite easy.

So first the rule itself. So the power rule states that the would D with a over DX of X to the end equals N times X to the power of N minus one here N can actually be any number. But in practice, we are going to just be looking at the case today for when it isn't when, and is a whole number. So the proof is as follows. We first write out the limit definition of deriv of the derivative. Next we expand out the term X plus H all to the power of all to the power of N into a partially expanded sum of terms. Now, we do it as, as follows. We write X to the N plus H times, N times X to the N minus one. And now for the rest of the terms, we aren't really concerned about terms with X of lower degree. So we just write H to the second times a where a is a standard for the other terms, minus X to the N now, okay. Now X to the end, a negative X to the end cancel.