Hey EconNor.

As a general rule, we find the inverse function (if it exists) by swapping the independent and dependent variable and re-solving for an expression of the new independent variable in terms of the dependent variable. In other words,

y = f(x) [Original function] Swapping gives us

x' = f(y').

Now re-arrange to get

y' = g(x') and g is our inverse function.

In your example y = x and x = n. Doing the above gives us:

n = Ax^e

(n/A)^(1/e) = x so our inverse function is

x^(-1)(n) = (n/A)^(1/e) just as you predicted.

For the second we use the normal power rule which gives:

d/dn[(n/A)^(1/e)]

= (1/A)^(1/e) * (1/e) * n^(1/e - 1)

Where did you get your w from?