Home » Blog » Find the derivative of arcsin(sin x – cos x)

Find the derivative of arcsin(sin x – cos x)

Find the derivative of the function

    \[ f(x) = \arcsin (\sin x - \cos x). \]


Using the formula for the derivative of the arcsine and the chain rule we have

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

This is valid for all x such that \sin (2x) \neq 0.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):