If f(x) = 1/x^3, what is the derivative of the inverse of f(x)?

To find the derivative of the inverse of a function, we can use the formula:

(f^-1)'(y) = 1 / f'(x)

where y = f(x). From the question, we have:

f(x) = 1 / x³.

First, we need to find the derivative of f(x):

f'(x) = -3 / x⁴.

Next, we need to express x in terms of y. Since y = f(x), we can write:

y = 1 / x³ => x³ = 1 / y => x = (1 / y)^1/3.

Now, we substitute x back into the derivative:

(f^-1)'(y) = 1 / f'(x)

(f^-1)'(y) = 1 / (-3 / x⁴) = -x⁴ / 3.

Finally, we substitute x = (1 / y)^1/3:

(f^-1)'(y) = -((1 / y)^1/3)⁴ / 3 = -1 / (3 * y^4/3).

So the derivative of the inverse function is:

(f^-1)'(y) = -1 / (3 * y^4/3).