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) `

निम्न फलनों का अवकलन गुणां

1.	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) `
Answer» 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 t3x^(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 `