107. Derivatives of Inverse Functions
Finding the derivative of the inverse function at a given point directly from the function . Instead of explicitly computing the inverse function , we use the inverse function derivative formula:
This approach allows us to determine the derivative of the inverse function without needing to express explicitly. Instead, we find the value of that satisfies (where a a is the given point), evaluate , and apply the formula.
1. Definition of Inverse Function
2. Differentiate Both Sides
Differenatiate both sides with respect to
The right-hand side simplifies to:
Using chain-rule:
The left side expands as:
This we get:
3. Solve for
Rearrange to isolate :
Example
Given the function:
We want to find at using the inverse function derivative formula:
1. Compute
Differenatiate :
2. Solve for such that
We need to find such that:
Solving for :
3. Compute
Evaluate the derivative at :
4. Use the formula
5. Interpretation
The expression:
represents the derivative of the inverse function evaluated at . This means it gives the slope of the tangent line to the inverse function at .
inverse.py
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)