Question

How does one solve y=x^(1/x) using Logarithmic Differentiation?

Answer 1

are you trying to find derivative of y ?

Answer 2

d/dx of the entire equation?

Answer 3

then you want to find \(\large \dfrac{dy}{dx}\)

Answer 4

What's the difference between d/dx vs. dy/dx?

Answer 5

take natural log both sides,
then differentiate both sides

Answer 6

I don't know how to :(

Answer 7

d/dx is a operator

dy/dx is the derivative of y

Answer 8

okay

Answer 9

Can you point me to a good free resource that'll teach me how to differentiate logarithmic functions?

Answer 10

see
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/derivatives-inverse-functions/v/calculus-derivative-of-x-x-x

Answer 11

ok thank you

Answer 12

\[y = x^{\frac{1}{x}}\]
\[\log~ y = \log~ (x^{\frac{1}{x}}) = \frac{1}{x} \log~x ~~~\because \log ~a^b = b~\log~a\]
\[\frac{d}{dx} [ \log~y] = \frac{d}{dx} [\frac{1}{x} \log~x]\]
Now differentiate. :)

Answer 13

Could you show me how to differentiate?

Answer 14

\[\frac{d}{dx}(\ln(x))=\frac{1}{x} \\ \frac{d}{dx}\ln(f(x))=\frac{1}{f(x)} \cdot f'(x)\]