Understanding the Derivative of the Inverse Function: A Step-by-Step Proof

derivative of inverse

The derivative of the inverse of a function is given by the formula:

d/dx (f^(-1)(x)) = 1 / (dy/dx)_x

Here, f^(-1)(x) represents the inverse function of f(x), and (dy/dx)_x represents the derivative of the original function f(x) evaluated at the point x

The derivative of the inverse of a function is given by the formula:

d/dx (f^(-1)(x)) = 1 / (dy/dx)_x

Here, f^(-1)(x) represents the inverse function of f(x), and (dy/dx)_x represents the derivative of the original function f(x) evaluated at the point x.

To prove this formula, we can use the definition of the derivative:

(d/dx (f^(-1)(x))) = lim (h -> 0) [(f^(-1)(x + h) – f^(-1)(x)) / h]

Let y = f^(-1)(x), then we have x = f(y). Now, let’s differentiate both sides with respect to x:

d/dx (x) = d/dx (f(y))

1 = dy/dx

Now, we can rewrite the expression from the definition of the derivative using y instead of f^(-1)(x):

(d/dx (f^(-1)(x))) = lim (h -> 0) [(y + h – y) / h]

Simplifying, we get:

(d/dx (f^(-1)(x))) = lim (h -> 0) (h / h)

(d/dx (f^(-1)(x))) = lim (h -> 0) 1

(d/dx (f^(-1)(x))) = 1

Thus, we have shown that the derivative of the inverse function is equal to 1, which matches the formula stated at the beginning.

More Answers: