Question

Verify Inverse function theorem for finding the derivative at a point y of the 0
domain of the inverse function of the function f (x) = cosx, x ∈[0,π].Hence, find
the derivative of the inverse function aty

Answer 1

Answer on Question #64863 – Math – Real Analysis

Question

Verify Inverse function theorem for finding the derivative at a point $y$ of the 0 domain of the inverse function of the function $f(x) = \cos x, x \in [0, \pi]$. Hence, find the derivative of the inverse function at $y$.

Solution

Derivative of the Inverse Function:

$$\frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}.$$ $$f(x) = \cos x, \quad f'(x) = -\sin x, \quad f^{-1}(x) = \arccos x.$$ $$y = f(x) = 0 \text{ when } x = \frac{\pi}{2}.$$

Thus

$$\frac{d}{dx} f^{-1}(x)|_{x=0} = \frac{1}{-\sin(\arccos(0))} = -\frac{1}{\sin\left(\frac{\pi}{2}\right)} = -1.$$

Answer: $-1$.

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