x के सापेक्ष निम्नलिखित का अवकलन कीजिए :
(i) ` cos ^(-1) ( sin x) ` (ii) ` tan^(-1) ((sinx)/(1+cosx))`
(iii) ` sin^(-1) ((2^(x+1))/(1+4^(x))).`

Explanation

(i) माना ` y = cos^(-1) (sin x) `
` = cos^(-1) [ cos^(-1) p cos ( 90^(@)-x) ]`
` y= 90^(@) - x` ,
` :. (dy)/(dx) = 0-1 = -1 `
(ii) माना ` y = tan^(-1) ( (sinx)/(1+cosx)) `
` = tan^(-1) ((2 sin* x/2cos .x/2)/(2 cos ^(2). x/2))`
` = tan^(-1) (tan .x/2)`
` = x/2`
` :. (dy)/(dx) = 1/2 `
(iii) ` y = sin^(1) ((2^(x+1))/(1+4^(x)))`
` = sin ^(-1) ((2^(x).2)/(1+(2 ^(x))^(2)))`
यहाँ` 2^(x) =tan theta` रखिए ।
` y = sin^(-1) ((2 tan theta)/(1+ tan theta ) )`
` sin ^(-1) ( sin 2 theta) `
` y = 2 theta = 2 tan^(-1)(2 ^(x)) `
` :. (dy)/(dx) = 2.(1)/(1+ (2^(x))^(2) ). d/(dx) 2^(x)`
` 2/(1+ 4^(x)) . 2 ^(x) log2`
` = (2 ^(x+1)log 2)/ (1+4)`