Question

If $f\left(x\right)=(x+1{)}^x+x^{(x+1)}$ then find the value of $f^{\prime }\left(1\right)$.

Answer 1

Given,

$f\left(x\right)=(x+1{)}^x+x^{(x+1)}$.

or, $f\left(x\right)=e^{\left(x\right)\log (x+1)}+$$e^{(x+1)\log \left(x\right)}$.

Now, $f^{\prime }\left(x\right)=(x+1{)}^x(\frac{x}{x+1}+\log (x+1\left)\right)+x^{(x+1)}(\frac{x+1}{x}+\log x)$......(1).

From (1) we have,

$f^{\prime }\left(1\right)=2(\frac{1}{2}+\log 2)+1(2+0)$

or, $f^{\prime }\left(1\right)=3+2\log 2$.