`sqrt(sin 3 x) ` का x के सापेक्ष प्रथम सिद्धांत द्वारा अवकल गुणक ज्ञात कीजिए ।

Explanation

` :' f(x) = sqrt( sin 3x)`
` :. f( x+h) = sqrt( sin 3(x+h))`
अब `(dy)/(dx) = lim _( h to 0) ( f( x+h) - f(x))/h`
` = lim_( h to 0) ( sqrt( sin 3( x+h))-sqrt( sin 3x))/h `
` lim _( h to 0) ( sin 3 ( x+ h) - sin 3x)/(h[sqrt( sin 3( x+h))+sqrt(sin 3x)])`
` = lim _( h to 0) ( 2 cos (3( x+h ) + 3x)/2.sin. (3(x+h) - 3x)/2)/(h[sqrt( sin 3 ( x+h))+ sqrt( sin 3x)])`
` = lim _( h to0) ( 2cos (3x+(3h)/2).sin .(3h)/2)/(h[sqrt(sin 3(x +h))+ sqrt(sin 3x)])`
` = lim_ ( h to 0) ( 2 cos (3x+(3h)/2). sin .(3h)/2 xx 3/2)/([ sqrt( sin 3 ( x+h))+ sqrt(sin 3x)]xx(3h)/2)`
` = lim _( h to 0) (2 cos ( 3x+ (3h)/2))/([sqrt(sin 3( x+h))+ sqrt( sin 3x)])xx lim_( x to 0) ((sin(3h)/2)/((3h)/2))xx3/2`
` :. (dy)/(dx) = ( 2cos 3x)/(2sqrt(sin 3x))xx1 xx 3/2`
` = ( 3 cos 3x)/( 2 sqrt( sin 3x))`