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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


(f^{-1})'(x)=\frac{1}{\sqrt{1-x^2}}

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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