To find the derivative of the inverse function, we first need to clarify a few concepts regarding inverse functions and derivatives.
Let’s denote the function as f(x) = x^(1/3). To find the derivative of the inverse function, we can utilize the relationship between a function and its inverse, given by the formula:
(f-1(y))’ = 1 / f'(x)
where f'(x) is the derivative of the function f evaluated at the point x, and y = f(x).
First, we calculate the derivative of f(x):
f'(x) = (1/3)x^(-2/3)
Now we need to find the inverse function of f. To do that, we set y = f(x) = x^(1/3). To solve for x in terms of y, we cube both sides:
y^3 = x
Thus, the inverse function is:
f-1(y) = y^3
Next, we need to evaluate the derivative of the inverse function:
(f-1(y))’ = 1 / f'(x), where x = f-1(y) = y^3 means we need to substitute x = y^3 into the expression for the derivative:
f'(y^3) = (1/3)(y^3)^(-2/3) = (1/3)(y^(-2)) = y^(-2)/3 = 1 / (3y^2)
Finally, substituting this back into the inverse derivative gives:
(f-1(y))’ = 1 / f'(y^3) = 1 / (1 / (3y^2)) = 3y^2
Therefore, the derivative of the inverse function of f(x) = x^(1/3) is:
(f-1(y))’ = 3y^2
In summary, the derivative of the inverse function of f(x) = x^(1/3) can be expressed as:
(f-1(y))’ = 3f(x)^2 when properly translated back to relate to original function variables.