`sin x^(2) ` का x के सापेक्ष प्रथम सिद्धांत से अवकल - गुणांक ज्ञात कीजिए ।

Explanation

माना ` f(x) = sin x^(2)`
तब `f ( x+h) = sin ( x + h)^(2)`
` d/(dx) sin x^(2) = lim _( h to 0 ) (sin( x+h)^(2)-sin x^(2))/h`
` = lim _( h to 0 ) ((2cos(x+h)^(2)+x^(2))/2.(sin (x+h^(2))-x^(2))/2)/h`
` lim _( h to 0 )((2cos(x+h)^(2)+x^(2))/2.sin(h+2x)/2)/h `
` lim _(h to 0 ) ((2cos(x+h)^(2)+x^(2))/2.sin . (h (h +2x))/2 xx ((h + 2x))/2)/(h (( h+2x)/2))`
` 2 lim _( h to 0 ) cos ((2x^(2) + 2hx + h^(2))/2)xx lim _(h to 0 ) ((sin. (h(h+2x))/2)/(h( h + 2x)/2))xx1/2 lim _ ( h to 0 ) ( h + 2x) `
` = 2x cos x^(2) `