Calculus Help

[MeklorIlavator]

My Calculus skills must have gotten really rusty over the summer, so I need some help doing a review. I have to differentiate the following problems:

A) f(x)={x^[sin(x)]}^[tan(x)]

B) f(x)=arctan(x)/{[3^x+ln(x)]^(1/2)

Now, for the first one I know that one needs to take the natural log of both sides to get the function down out out the exponent(it comes out as tan(x) * sin(x) * ln(x)), but after that I feel like I'm forgetting something. Or should I just go onto using the product rule?

As to the second, I have no clue how to proceed except by the division rule.

Am I missing anything?

[Tirian]

I think you're leaving a critical set of brackets out of the first problem. (A^B)^C is different from A^(B^C).

[RandomLogic]

Its been a while for calculus for me as well but for #2 I would just use the product rule like you think. (hodihi-hidiho)/(hoho)

[MeklorIlavator]

Very true, they should be fixed now. Of course, I think that means my work is wrong now. Oh well.

[Tirian]

No, I think you're on the right track. f(x) = (x^(sin x))^(tan x) = x^(sin x tan x) = x^(sin^2 x / cos x). Then, like you said, ln f(x) = (ln x) ln(sin^2 x / cos x) and you take the differential of both sides, propogate the chain rule like mad, and then solve for f'(x). The other bracketing would have been considerably uglier, I suspect.

[Jorkens]

Its been a while for calculus for me as well but for #2 I would just use the product rule like you think. (hodihi-hidiho)/(hoho)
Yeah, product rule or quotient rule (depending on whether you look at it as a/ (b^{1/2}) or a*b^{-1/2} ) and then do each bit by using the chain rule.

And yes, agree with Tirian on the second one. Use the chain rule on log(f(x)) (and that and a lot of product rule on the right hand side) and use your original expression for f(x) to get rid of any f(x)s that are left hanging around.