Question

derivative of arcsin(cos(2x))

Answer 1

Let
\[f(x)=\arcsin\left(\cos(2x)\right)\]
So :
\[f'(x)=-2\sin(2x)\times \frac{1}{\sqrt{1-\cos^2(2x)}}\]

Answer 2

i would do something slightly different

Answer 3

or just look more carefully at what you have in the answer above

Answer 4

since \(1-\cos^2(2x)=\sin^2(2x)\) the answer given above is \[f'(x)=-2\sin(2x)\times \frac{1}{\sqrt{1-\cos^2(2x)}}\]
\[-2\sin(2x)\times \frac{1}{\sqrt{\sin^2(2x)}}=-\frac{\sin(2x)}{|\sin(2x)}\]

Answer 5

typo there, i forgot the 2 in the numerator
the point is that the derivative is either 2 or -2 depending on the "sign" of \(\sin(2x)\)