Question

Derivative of ( cos( 2 sin inverse of cosx))

Answer 1

$\huge \texttt{ \red{ \underline{question }}}\: \: \\ \\ \small \bf{ find \: the \: derivative \: of \: \cos \big(2 { \sin}^{ - 1} \cos(x) \big) } \\ \\ \\ \huge \: \texttt{ \underline{answer}} \\ let \: \\ y \: = \cos {\big(2 { \sin}^{ - 1} \cos(x) \big) } \:$

Differentiating on both sides with respect to "x",

$\small \frac{dy}{dx} = - \sin(2 { \sin }^{ - 1} \cos(x) ) \frac{d}{dx} \big(2 { \sin }^{ - 1} \cos(x) \big) \\ \\ \small \frac{ dy}{dx} = - \sin(2 { \sin }^{ - 1} \cos(x) ) \: \bigg( \frac{2}{ \sqrt{1 - { \cos }^{2} x} } \bigg) \frac{d}{dx} \cos(x) \\ \\ \small \: \boxed{ \frac{dy}{dx} = - \sin(2 { \sin }^{ - 1} \cos(x) ) \bigg( \frac{2}{ \sin(x) } \bigg) \times \big( - \sin(x) \big)}$

This is the required answer .