निम्न फलनों का अवकलन गुणांक ज्ञात कीजिए|
` (i) ((x-sin x )^(3//2))/( sqrt(x) ) " "(ii) sin (log x ) " "(iii) cos x^(3) " "(iv) e^(log_ax) `

Explanation

माना ` y= ((x-sin x )^(3//2)) /( sqrt(3)) `
` therefore " "(dy)/(dx) =(sqrt(x) (d)/(dx) (x-sin x ) ^(3/2) -(x-sin x)^(3/2) (d)/(dx) (x) ^(1/2))/(x)`
माना `x-sin x=t `
अतः `(dy)/(dx) =(sqrt(x) (d)/(dt) t^(3/2) (d)/(dx)(x-sin x) -(x-sin x )^(3/2)(d)/(dx) x^(1/2)/(x) `
` = (sqrt (x) *(3)/(2) t^(1//2) (1-cos x ) -(x-sin x)^(3//2) (1)/(2sqrt(x)) )/(x) `
` " "= (3xsqrt (x-sin )(1-cos x)-(x-sin x)^(3//2))/(2xsqrt x))`
` sqrt ((x-sin x) [3x (1-cos x-(x- sin x ) ]))/(2xsqrt (x))`
` =(sqrt ((x-sin x))[ 2x -3cos x+ sin x] )/(2xsqrt (x))`
माना `y= sin (log x ),`माना `log x =t`
` therefore " "y= sin t `
` therefore (dy)/(dx) =(d)/(dt) sin t (d)/(dx) log x=(cost )/(x) =(cos (log x ) )/(x) `
(iii) माना ` y cos x^(3),` माना ` x^(3) =t `
` " " y= cos t `
` therefore " "(dy)/(dx) =(d)/(dt) cos t (d)/(dx) x^(3) =-sin t*3x^(2) `
` therefore (dy)/(dt) =- 3x ^(2) sin x^(3) `
(iv) माना ` y= e ^(log _ax) , ` माना ` log _a x=t `
` therefore " "y= e^(t) `
` therefore (dy)/(dx) =(d)/(dt) e^(t) (d)/(dx) log_a x `
` " "= e ^(t) (1)/(x) log_a e= e^(log _a x) (1)/(x) log_ae `