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Problem 29 Hard Difficulty

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

$ f(x) = \dfrac{1}{x} $ , $ \dfrac{f(x) - f(a)}{x-a} $


$-\frac{1}{a x}$

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Video Transcript

we're going to evaluate this difference quotient for the function F of X equals one over X notice that we need to know f of a so f of a is going to be one over a. Okay, so now let's substitute F of X one over X into our quotient and f of a one over a into our quotient, and that's over X minus a. So here's the question that we're hoping to simplify. And any time you have a complex fraction like this where you have fractions inside your fraction, it's a good idea to multiply by a special form of one to clear those fractions. So I'm going to multiply the top and bottom by X times. A. I'm just gonna put that over one so that everything lines up nicely. So what happens then is when we multiply one over X by X times a The X's cancel and we just have a minus. Now we multiply one over a by X Times A and the A's council, and we just have X. And then that's over. X minus a times X times a Nothing councils there, so I'm just going to call that X a times the quantity X minus a. Okay, let's look at what we have now. Notice that we have a minus X on the top and we have X minus A on the bottom. Those are opposites. And we can cancel opposites if we leave negative one. Just like we can cancel things that are equivalent if we leave positive one or the other way to think about it is just factor a negative one out of the numerator and it will be negative one times a quantity X minus a And now you can just cancel your X minus ace. So what do we have now? We have negative one over X times A. You could call it eight times X or x times a.