Question

First and second derivatives of (x+1)^(x+1) . i've tried wolfram but didn't understand the steps ... specially for the second derivative

Answer 1

isn't (x+1)^(x+1)=(x+1)(x+1)^x

or am i wrong ?

Answer 2

you can't do that

Answer 3

use \[f(x)=e^{\ln(f(x))}\]

Answer 4

(x+1)^(x+1)=e^[(x+1)ln(x+1)]

Answer 5

i don't know that rule .. where can i found the proof of it ?

Answer 6

e and ln are inverse functions

Answer 7

\[f^{-1}(f(g(x))=g(x)\]

Answer 8

To find the first derivative, use logarithmic differentiation. Take the expression:
\[y = (x+1)^{x+1} \]
and take the logarithm of both sides:
\[\ln |y| = \ln |(x+1)^{x+1}| \]
Now use implicit differentiation to find \[\frac{dy}{dx}\]which is the desired result.
For that, use the following rule:
\[\frac{d}{dx}(\ln|x|)=\frac{1}{x}\]

Answer 9

Notice that \[\ln|y|\] is the natural logarithm of the absolute value of y (between two vertical bars: |y|).