How to prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$ Prove that there is a differentiable function $f$ such that $[f(x)]^{153}+f(x)+x=0$ for all $x$. Furthermore, find $f'$ in terms of $f$.
To me I just write $y$ instead of all $f(x)$ and find that $x=-(y^{153}+y)$ If I just derive that would that be the solution of this question? If not what should i do to answer it appropriately.

HINT:  
If $x = -[f(x)]^{153} - f(x)$,
then $g(x) = -x^{153} - x$ is the inverse function of $f(x)$.
Then if you prove $g(x)$ is differentiable, what does it tell about its inverse function ?
