derivative of inverse trig: show f(x)=arcsin((x-2)/2)-2arcsi
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skeeter
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\(\displaystyle f(x) = \arcsin(u) - 2\arcsin(v)\)
\(\displaystyle u = \frac{x-2}{2}\)
\(\displaystyle u^2 = \frac{(x-2)^2}{4}\)
\(\displaystyle u' = \frac{1}{2}\)
\(\displaystyle v = \frac{\sqrt{x}}{2}\)
\(\displaystyle v^2 = \frac{x}{4}\)
\(\displaystyle v' = \frac{1}{4\sqrt{x}}\)
\(\displaystyle f'(x) = \frac{u'}{\sqrt{1 - u^2}} - \frac{2v'}{\sqrt{1 - v^2}}\)
sub in the values and do the grunt work algebra ... you should get \(\displaystyle f'(x) = 0\)
\(\displaystyle u = \frac{x-2}{2}\)
\(\displaystyle u^2 = \frac{(x-2)^2}{4}\)
\(\displaystyle u' = \frac{1}{2}\)
\(\displaystyle v = \frac{\sqrt{x}}{2}\)
\(\displaystyle v^2 = \frac{x}{4}\)
\(\displaystyle v' = \frac{1}{4\sqrt{x}}\)
\(\displaystyle f'(x) = \frac{u'}{\sqrt{1 - u^2}} - \frac{2v'}{\sqrt{1 - v^2}}\)
sub in the values and do the grunt work algebra ... you should get \(\displaystyle f'(x) = 0\)