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

Homework Equations

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)

 

[Werg22]

Ridiculously boring. Who gives you those exercises?

 

[JasonRox]

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?

 

[cristo]

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!

 

[HallsofIvy]

By the way, for [a] y=ln(x^4 sin^2 x)
I would write y= 4 ln x+ 2 ln sin x and get
$$y'= \frac{4}{x}+ 2\frac{cos x}{sin x}$$
Can you show that is the same as your answer?