Question

use logarithmic differentiation to find y = x^(lnx)

Answer 1

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

Answer 2

you have a choice. myininaya will do it one way, i will do it another

Answer 3

\[\ln(y)=\ln(x) \ln(x)\]

Answer 4

\[\ln(y)=[\ln(x)]^2\]

Answer 5

\[\frac{y'}{y}=2 \ln(x) \frac{1}{x}\]

Answer 6

now multiply y on both sides

Answer 7

myininya did u do product rule?

Answer 8

chain rule

Answer 9

\[x^{\ln(x)}=e^{\ln(x)\times \ln(x)}\]so
\[\frac{d}{dx}x^{\ln(x)}=\frac{d}{dx}e^{\ln(x)\times \ln(x)}\]
\[=e^{\ln(x)\times \ln(x)}\times \frac{d}{dx}\ln(x)\times \ln(x)\] and
\[\frac{d}{dx}\ln^2(x)=2\ln(x)\frac{1}{x}\] so your final answer is
\[=e^{\ln(x)\times \ln(x)}\times \frac{2\ln(x)}{x}\]

Answer 10

why did u multiply by 1/x?

Answer 11

chain rule. the derivative of
\[f^2(x)\] is
\[2f(x)f'(x)\]

Answer 12

\[\frac{d}{dx}\ln^2(x)=2\ln(x)\ln'(x)=2\ln(x)\frac{1}{x}\]

Answer 13

thamk you!

Answer 14

when you see
\[f(x)^{g(x)}\] and you want the derivative, you have a choice. you can
1) take the log
2) take the derivative of the log
3) multiply the result by the original function

Answer 15

or, you can rewrite
\[f(x)^{g(x)}=e^{g(x)\ln(f(x))}\] and take the derivative using the chain rule

Answer 16

it amounts to the same thing because all the work is finding the derivative of
\[g(x)\ln(f(x))\]