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Find $ f'(a) $.

$ f(t) = 2t^3 + t $

$f^{\prime}(a)=6 a^{2}+1$

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Missouri State University

Harvey Mudd College

Idaho State University

So in this problem we are given the function F. Of X equals two x cubed plus X. We were asked to determine if prime at a. Okay well by definition F. Prime at A. Is the limit As a church goes to zero of F. Of a plus H minus F. Of a. All over H. Okay so first of all we got to figure out half of a plus H. Well that's two times A. Plus H cubed plus A plus H. So that's two times E cubed was three A squared H. Plus three A. H squared plus one O. C. M. P. H. Sorry And that's plus A. Plus H. And so this becomes, well I have to a cute and I have three A squared H. And I have three A. That's not three. Sorry that's what babies and this one This will be six and that would be six. And then plus and a. Okay this is H cubed over here was and each plus an h cubed wouldn't it? Okay and let me just make this a little clear. Sorry about that. Just make this clear. That's too H cubed plus a. We'll see each. Okay F it a is simply to a cubed was hey isn't it? Okay so that means that F. At A plus h minus F. A. R. Numerator up here. Right our numerator up here. Well let's see The two A Cubes are going to cancel out. I'm going to have six A squared H. Plus six A. H plus two H cubed. Hey minus A. That will be gone. So I'll be plus H. Left on the end and when it, okay so that means now that I'm left with F. Prime at A. Is the limit as H. goes to zero Of six A. Squared H. Plus six h. plus two H. Cubed plus H. All over H. And now what do I see that age cancels? And that one cancels one of those cancels and that cancels doesn't it? Okay so yes I'm now left with this is the limit His age goes to zero of six A squared plus six A. Plus two eight squared. Well not mess up up here something didn't work out right I left two squared off of that turn right there didn't I? So that one squared. So this one was squared and so it should have been and H. Left there. Let's fix that up. This is the age squared cancel one of those out. Okay So this is six. Let's fix this up. Six A. Aah plus two H. Squared. Um So when I canceled this All right here that would have left or one. Okay so this is plus one down here. All right so now H. Goes to zero. So a return has an ancient it goes to zero. This one goes to zero and that one goes to zero So much with six A. Squared plus one is F. Prime today

Oklahoma State University