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Evaluate the difference quotient for the given function. Simplify your answer.

$ f(x) = x^3 $ , $ \dfrac{f(a + h) - f(a)}{h} $

$3 a^{2}+3 a h+h^{2}$

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all right, we're going to evaluate this difference quotient for the function F of X equals X cubed. And this kind of problem is really good practice for something that's coming up in another chapter or two. So we're going to need to find f of A plus h. Let's do that off to the side F of a plus age would be a plus age quantity cubed. We're also going to need to find 1/2 of a after they would be a cubed. So let's go ahead and substitute those into the quotient. And we have a plus age quantity cubed minus a cubed over h. Okay, now the bulk of the work is going to be figuring out how to expand a plus h quantity. Cute. There is a shortcut for this. If you happen to know the binomial theorem and you know how to use Pascal's triangle as your coefficients, you'll get one A to the third power plus three, a squared age plus three, a age squared, plus one age cubed. If you don't know that binomial theorem shortcut, what you're going to do is multiply a plus h times a plus h times a plus H, and you can do that by multiplying two of them together using the foil method. And when you're done with that, then you multiply it by the third, so you get a squared plus two a h plus H squared. Now you multiply it by the 3rd 1 And when you're done with all of that and you have combined the like terms, it's going to end up to be equivalent to what I have above. Okay, so expanding the try the binomial cubed gives us a cube plus three a squared age plus three a age squared minus a plus. H Q. Excuse me and then we still have minus a cubed at the end, and that's all over. H notice that we haven't a cube Jenna minus a cubed. Let's cancel those and let's look at what we have left. All three terms in the numerator have a factor of H, so if we factor h out of them, the other factor will be three. A squared plus three a h plus h squared, and that's all over H. So we can cancel the age from the top of the bottom. And what we have now is three a squared plus three age plus age squared