107. Derivatives of Inverse Functions

Finding the derivative of the inverse function ${𝑓}^{-1}\left(𝑥\right)$ at a given point directly from the function $𝑓\left(𝑥\right)$. Instead of explicitly computing the inverse function ${𝑓}^{-1}\left(𝑥\right)$, we use the inverse function derivative formula:

This approach allows us to determine the derivative of the inverse function without needing to express ${𝑓}^{-1}\left(𝑥\right)$ explicitly. Instead, we find the value of $𝑥$ that satisfies $𝑓\left(𝑥\right)=𝑎$ (where a a is the given point), evaluate ${𝑓}^{\prime }\left(𝑥\right)$, and apply the formula.

1. Definition of Inverse Function

2. Differentiate Both Sides

Differenatiate both sides with respect to $𝑥$

The right-hand side simplifies to:

Using chain-rule:

The left side expands as:

This we get:

3. Solve for $\frac{𝑑}{𝑑𝑥}\left[{𝑓}^{-1}\left(𝑥\right)\right]$

Rearrange to isolate $\frac{𝑑}{𝑑𝑥}\left[{𝑓}^{-1}\left(𝑥\right)\right]$:

from sympy import symbols, solve

# y = x**3 + x

x, y = symbols('x y')

f = x**3 + x - y

inverse = solve(f, x)

print(inverse)

107.1. Derivative Inverse Sin

107.2. Derivative Inverse Cos

107.3. Derivative Inverse Tan