Thread: Some derivatives problems

Hi everyone, I'm just study Calculus about two weeks. Now I got some practice exercises, I already done all of them except those following few questions in the attachment. I've got no idea how to do those questions at all
Can anyone help me a bit? thank you so much

Hey Carlos4662.

Guide us through your attempt so we can give you some specific feedback and pointers.

Hint: The first one involves a few applications so you need to use the chain rule twice, by doing d/dx f(g(u)) where u = h(x) and then using then using dg/dx = dg/du * du/dx.

Thanks for your reply. For your hint, maybe I missed something at my lessons. I only know about the chain rule, the following steps that you mentioned I've totally no idea about them. Can you give me more hint or show me some examples? Thank you so much.

The first is trying to force you to see how the chain rule potentially (kind of) iterates, or nests.

Just in case a picture helps...



Try to fill the blanks?

Full size

For the second you can use the log differentiation method (just as you will also tackle the third problem)...



Alternatively, use the triple product rule (see Product rule - Wikipedia, the free encyclopedia)...



For the last problem, do implicit differentiation...



... then solve for dw/dx, and differentiate again (either directly, or implicitly).


Key in spoiler ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

And...



... is the product rule, where, again, straight continuous lines are differentiating downwards with respect to x.

_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

thank you so much for your reply and pictures. Problems solved.