I know that the Fenchel conjugate of a function is $$f^*(x^*) = \sup_x\{\langle x, x^*\rangle - f(x)\}.$$ However, how do I find the Fenchel conjugate of the function $$f(x) = \frac{1}{p}\sum\limits_{i=1}^n |x_i|^p$$ where $1 < p < \infty$.

I have tried differentiating the equation and taking it to be $= 0$ but I cannot seem to reach any answer.