The geometric approach

Suppose that $f:(0,\infty)\rightarrow(0,\infty)$ is given by f(x) = x^p where p is an integer larger than 1. Then $g:(0,\infty)\rightarrow(0,\infty)$ given by g(x) = x^1/p is the inverse of f. Suppose we want to find the slope of the tangent line to y=g(x) at the point (a, a^1/p). As we have argued before, the slope of this line is the reciprocal of the slope of the tangent line to y = f(x) at the point (a^1/p,a), and the slope of this tangent line is given by f'(a^1/p). Since f'(x) = px^p-1 we have f'(a^1/p) = pa^1-(1/p), and so the slope of the tangent line to y = g(x) at (a,a^1/p) is (1/p)^(1/p)-1, which agrees with our algebraic computation.