निम्न फलनों का अवकलन गुणांक ज्ञात कीजिए-
` (i)log [ x + sqrt (x^(2) -a ^(2)) ]`
` (ii) log sin (ax+ b) `
`(iii) sin [ cos (2x+ 3) ]
` (iv) log [ sin mx + cos nx ]`

Explanation

माना ` y= log [ x + sqrt (x^(2) -a^(2) ) ] `
माना ` x+ sqrt (x^(2) -a^(2) ) =t `
` " " y = log t `
` therefore " "(dy)/(dx) =(d)/(dt) log t* (d)/(dx) [ x+ sqrt( x^(2) -a^(2)]`
` " "= ( 1) /(t) [ (d)/(dx)x+ (d)/(dx) (x^(2) -a^(2) ) ^(1//2) ] `
पुनः ` x^(2)-a^(2) =t_1 `
` therefore (dy)/(dx) =(1)/(t) [1+ (d)/(dt_1) t_1^(1//2) (d)/(dx) (x^(2) -a^(2) ) ]`
` " "= (1)/(t) [ 1+ (1)/( 2sqrt ( t_1) )(2x-0 ) ] =(1)/(t) [ 1+ (x)/(sqrt( x^(2) -a^(2)))] `
` " "= [ (1) /(x+ sqrt (x^(2) -a^(2) ) )] [ (x+ sqrt(x^(2) -a^(2) )) /( sqrt( ( x^(2) -a^(2)) ) ]] `
` " "= (1) /(sqrt( x^(2) -a^(2) ) ) `
माना ` y= log sin ( ax+ b) `
माना ` sin (ax+b) =t `
` therefore " " y= log t `
` therefore " "(dy)/(dx) =(d)/(dt) log t (d)/( dx) sin (ax+b) `
` " "= (1)/(t) (d)/(dt) sin ""t_ 1 (d)/(dx) (ax+ b) `
जबकि ` ax+ b =t_1 =(1)/(t) cos t_1 *(a+0) `
` " "(acos (ax+ b))/(sin (ax+ b) ) =a cot (ax+b) `
(iii) माना ` y= sin [cos (2x+ 3) ]`
माना ` cos (2x+ 3) =t `
` therefore " "y= sin t `
` therefore " "(dy)/(dx) =(d)/(dt ) sin t (d)/(dx) cos (2x+ 3) `
` " "=cost (d)/(dt_1) cos t_1"" (d)/(dx) (2x+ 3) `
जबकि ` (2x+3) =t_1`
` " "=- sin t_1 cost*2`
` =- 2cos [ cos (2x+ 3) sin (2x+ 3) ]`
(iv) माना ` y= log [ sin mx + cos nx ]`
माना ` sin mx + cos nx =t `
` therefore " " y= log t `
` therefore " "(dy)/(dx) =(d)/(dt) log t (d)/(dx ) [ sin mx+ cos nx]`
` " "= ( 1)/(t) (mcos mx -n sin nx ) `
` " "= (mcos mx - n sin nx ) /( sin mx +cos nx ) `