What Is the Power Rule of Calculus?
So to expand out the term we're going to use the
Hi, my name is Bassem Saad, I'm a Math Ph.D. candidate at U.C. Davis, and I'm here today for About.com to answer the question: “What is the Power Rule for Calculus?”The Power Rule lets us take the derivative of x to the n without the use of limits. Simply drop the n in front of the x and place n minus one where the exponent used to be.For example, if we want to take the derivative of x to the fifth, the power rule just tells us to drop this five in front so it becomes a coefficient of x, and we put one minus five, or four, in the exponent.So now we know the derivative of x to the fifth is just five times x to the fourth. But we still need to prove that the Power Rule gives us the derivative of x to the n.The first step is to apply the definition of the derivative of x to the n. That is, we take the derivative of x to the n -- is the same thing as saying we want to take the limit as h goes to zero of the difference of x to the n, minus x to the minus h to the n, all over h.For the next step, we'll have to expand out this term, because as it's written we can't cancel out this h in the denominator, and we need to in order to the take the limit as h goes to zero.So to expand out this term we're going to use the Binomial Theorem. So we re-write our equation down here; that is, the limit as h goes to zero. We copy down the first term: x to the n, minus x to the n minus x to the n minus one times h.And now we can write out the rest of the terms from the binomial equation, but they'll all drop out as we'll see later, so it's easier just to write it as a sum. Plus the sum of x to the n minus i, ai, hi, as i starts from two to n.And this whole formula is divided by h. So every term in here, there is an h of degree two or higher. Here's a clearer look at the equation. So in the third step we're going to cancel all the terms that can be cancelled and get an equivalent expression. So we'll have it equal to the limit as h goes to zero. Notice, first of all, that the x to the n cancels out with the x to the n. Every term that's left on the numerator will have a multiplication of h to some power, or just h by itself.So now h in the denominator can cancel with all these other terms. What you're left with is n times x to the n minus one, plus the summation from i equals two all the way up to n, all your coefficients, x to the n minus i, and your h to the i minus one. For step number four we can take the limit as h equals zero, because this expression is continuous at h equals zero. That means all we have to do is plug in zero everywhere we see h. What we have left is n times x to the n minus one.Thus, we've just proven the Power Rule. Thanks for watching, and to learn more, visit us on the web at About.com.