Math Problem Statement
What's the derivative of f^−1(x)
Solution
The derivative of the inverse of a function , denoted , is given by the following formula:
This formula applies under the assumption that is a one-to-one (bijective) function and that , the derivative of , is non-zero at the relevant points. Here’s the step-by-step reasoning:
- Suppose , which implies .
- Taking the derivative of both sides with respect to , we use implicit differentiation:
- By the chain rule, this becomes:
- Solving for , we get:
This gives the derivative of the inverse function.
Would you like more details or have any questions?
Here are some related questions to further your understanding:
- How do you find the derivative of an inverse trig function?
- Can the formula for derivative be used for all functions?
- What happens if at any point?
- How do you compute the derivative of an inverse function for a given example?
- How is implicit differentiation related to inverse functions?
Tip: Make sure that is one-to-one (bijective) before finding the derivative of its inverse!
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Derivatives
Implicit Differentiation
Formulas
Derivative of inverse function: d/dx(f^−1(x)) = 1 / f'(f^−1(x))
Theorems
Inverse Function Theorem
Suitable Grade Level
Undergraduate (Calculus I)
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