Find the slope of the tangent line on f at x=1

[anneheather98]

So I'm studying for my first calculus exam. I've been stuck on this problem for hours now.

So I used this formula: lim h->0 f(x+h)-f(x) / h


After evaluating all this out and trying to remove the radical on the top it becomes a giant mess. I'm not sure what I'm doing wrong, I don't think this problem is supposed to take so incredibly long. If anybody could help me I'd appreciate it a lot. Thanks!

 

[tkhunny]

So I'm studying for my first calculus exam. I've been stuck on this problem for hours now.
View attachment 10265
So I used this formula: lim h->0 f(x+h)-f(x) / h


After evaluating all this out and trying to remove the radical on the top it becomes a giant mess. I'm not sure what I'm doing wrong, I don't think this problem is supposed to take so incredibly long. If anybody could help me I'd appreciate it a lot. Thanks!

I hope you did not use that formula. That would be totally wrong.

Try this one: lim h->0 [f(x+h)-f(x)] / h

 

[Dr.Peterson]

So I'm studying for my first calculus exam. I've been stuck on this problem for hours now.
View attachment 10265
So I used this formula: lim h->0 [f(x+h)-f(x)] / h


After evaluating all this out and trying to remove the radical on the top it becomes a giant mess. I'm not sure what I'm doing wrong, I don't think this problem is supposed to take so incredibly long. If anybody could help me I'd appreciate it a lot. Thanks!

We generally only use the definition of the derivative directly, when there is a good reason to (such as when we have no other way, or when a teacher requires it on an exam). Were you explicitly told to use the definition? Have you learned the product rule and other properties of derivatives that can be used to do this? If the answers are "no" and "yes", then I would use the product rule.

There is a reason we move beyond the definition. By first proving the various rules, and then using them to find derivatives, we save an immense amount of unnecessary work.

 

[Dr.Peterson]

To supplement what I said earlier, if you are in fact required to use the definition directly, it can be done without too much work, if you know (a) the trick on which the proof of the product rule is based, (b) the trick for proving the derivative of the exponential, and (c) the trick for proving the derivative of a radical. That is, your work essentially will duplicate three proofs you should have seen; this, again, is why we used proved theorems rather than "reinventing the wheel" (or rather, reconstructing the wheel) every time by always starting from scratch.

But I did it (since I do know the tricks), and it didn't take too long. If you want help with your efforts, we can do so, but we'll want to see your work in order to find your errors. The reason for this is in our guideline summary, which you should read if you haven't yet.