(x1^p+x2^p)^1/p
ok, lets think of a simpler example. say you had $\displaystyle (x^5 + 4)^4$, and you wanted to differentiate that with respect to $\displaystyle x$. how would you do it?
do the same thing here. when you want to find the derivative with respect to $\displaystyle x_1$, say, just imagine that everything else is a constant, and differentiate as you normally would
fine. lets say "4" was "y" in the same question. so we want to differentiate $\displaystyle (x^5 + 4)^y$ with respect to $\displaystyle x$. just follow the rules. the power rule says, bring the power down, multiply, and subtract one from the power as long as it's a constant. so do that. the derivative is
$\displaystyle y(x^5 + 4)^{y - 1} \cdot 5x^4$
see? can you apply this to what you are doing?
the power is NOT a variable! that's what i'm trying to get across to you. when taking partial derivatives, EVERY variable but the variable you are differentiating with respect to is treated as a constant. here i was differentiating with respect to "x", so "y" was treated as a constant, because it is not "x"
maybe "y" was a bad letter to use, because it is usually used to mean "a function of x". that is not how it is meant here. it is just some other variable that is treated as a constant when differentiating with respect to x
so now can you try the problem to make sure you get it