Find the derivative. (d)/(dx) \left[ \sin^(-1)(\cos e^x)-\cos^(-1)(\sin e^x) \right]

Question

Find the derivative.


`(d)/(dx) \left[ \sin^(-1)(\cos e^x)-\cos^(-1)(\sin e^x) \right]`

Answer 1

`\qquad\qquad\huge\underline{{\sf Answer}}`

`\qquad \tt \dashrightarrow \:(d)/(dx) \left[ \sin^(-1)(\cos e^x)-\cos^(-1)(\sin e^x) \right]`

`\qquad \tt \dashrightarrow \:(d)/(dx) \sin^(-1)(\cos e^x) - (d)/(dx)\cos^(-1)(\sin e^x)`

`\qquad \tt \dashrightarrow \: \bigg(\frac{1}{ \sqrt{1 - ( \cos( {e}^(x) ) ) {}^(2) } } * - \sin( {e}^(x) ) * {e}^(x) \bigg) - \bigg( - \frac{1}{ \sqrt{1 - ( \sin( {e}^(x) )) {}^(2) } } * \cos( {e}^(x) ) * {e}^(x) \bigg)`

`\qquad \tt \dashrightarrow \: \bigg(\frac{ - {e}^(x) \sin( {e}^(x) ) }{ \sqrt{1 - ( \cos( {e}^(x) ) ) {}^(2) } } \bigg) - \bigg( - \frac{ {e}^(x) \cos( {e}^(x) ) }{ \sqrt{1 - ( \sin( {e}^(x) )) {}^(2) } } \bigg)`

`\qquad \tt \dashrightarrow \: \frac{ - {e}^(x) \sin( {e}^(x) ) }{ \sqrt{1 - \cos {}^(2) ( {e}^(x) ) {}^{} } } + \frac{ {e}^(x) \cos( {e}^(x) ) }{ \sqrt{1 - \sin {}^(2) ( {e}^(x) ) {}^{} } }`

`\qquad \tt \dashrightarrow \: \frac{ - {e}^(x) \sin( {e}^(x) ) }{ \sqrt{ { \sin}^(2) ( {e}^(x) ) {}^{} } } + \frac{ {e}^(x) \cos( {e}^(x) ) }{ \sqrt{ \cos {}^(2) ( {e}^(x) ) {}^{} } }`

`\qquad \tt \dashrightarrow \: \frac{ - {e}^(x) \sin( {e}^(x) ) }{ { { \sin}^{} ( {e}^(x) ) {}^{} } } + \frac{ {e}^(x) \cos( {e}^(x) ) }{ { \cos {}^{} ( {e}^(x) ) {}^{} } }`

`\qquad \tt \dashrightarrow \: - {e}^(x) + {e}^(x)`

`\qquad \tt \dashrightarrow \:0`

I hope this helps, if you find any problem with steps or got a mistake in my explanation then feel free to ask me ~