Question

Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given. $$f^{\prime}(x)=1+[f(x)]^{2}$$

Name: Problem 46, Chapter 7: Calculus: One Variable
Uploaded: 2021-11-04T15:33:39-08:00
Duration: 1 min 46 s
Channel: Nick Johnson
Description: Find a formula for (f^-1)^'(x) given that f is one-to-one and its derivative satisfies the equation given. f^'(x)=1+[f(x)]^2

   Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given.
$$f^{\prime}(x)=1+[f(x)]^{2}$$

Calculus: One Variable

Garret J. Etgen,… 10th Edition

Chapter 7, Problem 46

Instant Answer

Solved by Expert Nick Johnson

11/04/2021

Step 1

Now, we can differentiate both sides with respect to $x$ using the chain rule: $$\frac{d}{dx} [f^{-1}(f(x))] = \frac{d}{dx} [x]$$ $$\left(f^{-1}\right)^{\prime}(f(x)) \cdot f^{\prime}(x) = 1$$ Now, we want to find a formula for Show more…

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Find a formula for $\left(f^{-1}\right)^{\prime}(x)$ given that $f$ is one-to-one and its derivative satisfies the equation given. $$f^{\prime}(x)=1+[f(x)]^{2}$$

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Key Concepts

Inverse Function Theorem

The Inverse Function Theorem states that if a function is one-to-one and its derivative is non-zero at a point, the function has a locally defined inverse that is differentiable. This theorem leads to the important formula (f?¹)'(x) = 1 / f'(f?¹(x)), which expresses the derivative of the inverse function in terms of the derivative of the original function.

Chain Rule

The Chain Rule is a key principle in calculus for differentiating composite functions. It is used in deriving the inverse function derivative formula by differentiating f(f?¹(x)) with respect to x, resulting in the relation 1 = f'(f?¹(x)) · (f?¹)'(x), from which the reciprocal formula (f?¹)'(x) = 1 / f'(f?¹(x)) is obtained.

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