# I just need my work checked, derivatives

## 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)

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Ridiculously boring. Who gives you those exercises?

JasonRox
Homework Helper
Gold Member
Ridiculously boring. Who gives you those exercises?
Professors who KNOW that their students are doing derivatives for the first time.

cristo
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.
Yes, presuming that the last question is asking for the derivative wrt x.

Great, thank you!

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