# I just need my work checked, derivatives

FlipStyle1308

## Homework Statement

I just need to find the derivatives of the following:
[a] y=ln(x^4 sin^2 x)
f(x) = x^2 lnx, also find f'(1)
[c] x^y = y^x

See above

## The Attempt at a Solution

[a] y' = [(4x^3)(sin^2x) + (x^4)(2sinxcosx)]/(x^4sin^2x) = (4sinx + 2xcosx)/(xsinx)

f(x)' = (2x)(lnx) + (x^2)(1/x) = 2xlnx + x
f'(1) = 2(1)ln(1) + 1 = 2(0) + 1 = 1

[c] ln(x^y) = ln(y^x)
ylnx = xlny
d/dx ( ylnx) = d/dx (xlny)
(1)(y')(lnx) + (y)(1/x) = (1)(lny) + (x)(1/y)(y')
lnxy' + y/x = lny + xy'/y
lnxy' - xy'/y = lny = y/x
y'(lnx - x/y) = lny - y/x
y' = (lny - y/x)/(lnx - x/y)

## Answers and Replies

Werg22
Ridiculously boring. Who gives you those exercises?

Homework Helper
Gold Member
Ridiculously boring. Who gives you those exercises?

Professors who KNOW that their students are doing derivatives for the first time.

FlipStyle1308
LOL...are my answers correct?

Staff Emeritus
Professors who KNOW that their students are doing derivatives for the first time.
Good point; we all had to learn differentiation at some point. There's no need for such a pompous reply.
LOL...are my answers correct?
Yes, presuming that the last question is asking for the derivative wrt x.

FlipStyle1308
Great, thank you!

$$y'= \frac{4}{x}+ 2\frac{cos x}{sin x}$$