Calculating Derivative – Computing a derivative of an inverse function – Exercise 2076

Exercise

Find the derivative of the inverse of the following function:

$f(x)=\sin x$

Final Answer

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Solution

Given the function:

$f(x)=\sin x$

Its inverse function is

$f^{-1}(x)=\arcsin x$

We use the formula to find the derivative of the inverse function and get:

$(f^{-1})'(x)=(\arcsin x)'=$

$=\frac{1}{(\sin (\arcsin x))'}=$

$=\frac{1}{\cos (arcsin x)}=$

Using the following trigonometric identity:

$\cos x =\sqrt{1-\sin^2 x}$

we get:

$=\frac{1}{\sqrt{1-\sin^2 (\arcsin x)}}=$

$=\frac{1}{\sqrt{1-x^2}}$

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