Question

Use logarithmic differentiation to find the derivative of the function.

Answer 1

\[y=x^{\sin(x)}\]

Answer 2

\[y=x^{\sin(x)}\]

Answer 3

ok we take the log and get
\[\sin(x)\ln(x)\]

Answer 4

now take the derivative using product rule and get
\[\cos(x)\ln(x)+\frac{sin(x)}{x}\]\]now multiply by original function to get the answer:
\[x^{\sin(x)}(\cos(x)\ln(x)+\frac{\sin(x)}{x})\]

Answer 5

this is the answer...

Answer 6

why do u have to multiply by the original function

Answer 7

it was wrong!

Answer 8

really?

Answer 9

ok first you have to multiply by the original function because step one was to take the log

Answer 10

\[\frac{d}{dx}\ln(f(x))=\frac{f'(x)}{f(x)}\] so \[f'(x)=f(x)\times \frac{d}{dx}\ln(f(x))\]

Answer 11

in other words to take the derivative first take the log, then take the derivative, then multiply by the original function

Answer 12

you have
\[y=x^{sin(x)}\]take the log get
\[\ln(x^{\sin(x)})=\sin(x)\ln(x)\]so far i think this is right

Answer 13

now take the derivative of
\[\sin(x)\ln(x)\] using product rule. the derivative of sine is cosine, so first term could be
\[\cos(x)\ln(x)\] the derivative of \[\ln(x)\] is \[\frac{1}{x}\] so second term is
\[\frac{\sin(x)}{x}\] and i think the derivative of that product is
\[\cos(x)\ln(x)+\frac{\sin(x)}{x}\]

Answer 14

what is it supposed to be?

Answer 15

it is correct they just wanted sin(x)/x b4 cox x ln x

Answer 16

need help with another

Answer 17

y=(cox(9x))^x

Answer 18

ok well as long as the first one was right we can continue, but i think i posted a response to the second one. did you see it?

Answer 19

y=(cox(9x))^x is this like this...
y=(cos(9x))^x

Answer 20

http://openstudy.com/users/kehara15#/users/kehara15/updates/4e165f250b8bc22757460914

Answer 21

\[y=(\cos(9x))^{x}\]

Answer 22

same idea. take the log. simplify using property of log. take the derivative. multiply by original function

Answer 23

did you check previous post? i think all details are there

Answer 24

of course they might want you to write
\[\frac{9x\sin(9x)}{\cos(9x)}\] as
\[9x\tan(9x)\]

Answer 25

so what is the full answer

Answer 26

e^9xtan(9x)