If f(x) = x-1/3, What is the Derivative of the Inverse of f(x)?
We will be using the concept of inverse function to solve this.
Answer: If f(x) = x-1/3, then the Derivative of the Inverse of f(x) is −3x-4.
Let's solve this step by step to find the answer.
Given that, f(x) = x-1/3
Let y = f(x) = x-1/3
y = x-1/3
Cubing on both sides:
y3 = (x-1/3)3
y3 = x-1
x = 1/y3
x = y-3
We know from the definition of inverse function that: f(x) = y ⇒ x = f-1(y)
f-1(y) = y-3
Now change the variable from y → x,
∴ f-1(x) = x-3
d/dx (f-1(x)) = d/dx (x-3)
= −3 x-3 -1
= −3 x-4
Thus, if f(x) = x-1/3, then the derivative of the inverse of f(x) is −3x-4.