Exercise
Find the derivative of the inverse of the following function:
f(x)=\sin x
Final Answer
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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