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Find the derivative of arccos ((1-x)/21/2)

by RoRi

Find the derivative of the function

    \[ f(x) = \arccos \frac{1-x}{\sqrt{2}}. \]

Using the formula for the derivative of \arccos x and the chain rule we have

    \begin{align*} f'(x) &= \left( \frac{-1}{\sqrt{2}} \right) \left( \frac{-1}{\sqrt{1- \left( \frac{1-x}{\sqrt{2}} \right)^2}}\right) \\ &= \frac{1}{\sqrt{2 \left( 1 - \frac{1}{2} + x - \frac{x^2}{2} \right)}} \\ &= \frac{1}{\sqrt{1+2x-x^2}}. \end{align*}

Since the domain of \arccos x is |x| < 1, this formula is valid for |x-1| < \sqrt{2}.

One comment

  1. Artem says:

    I think there is a mistake in final statement in this, as well as other similar exercises: the domain of arccos(x) is actually |x| <= 1, not |x| < 1. It is that the DERIVATIVE exists only for |x| < 1. So, the derivative does not exist at the endpoints of the arccos domain.

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