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

Explanation

`f(x)= cos^(2)x^(2)`
` f ( x +h) = cos^(2) ( x + h)^(2)`
` d/(dx) cos^(2) x^(2) = lim _( h to 0 ) ( cos^(2) ( x+h)- cos^(2)x^(2))/h`
`= lim _( h to 0 ) ( cos^(2) ( x^(2)+h^(2)+2xh)-cos^(2)x^(2))/h`
` = lim _( h to 0 ) ( - sin ( 2x^(2)+h^(2)+2xh)sin (h^(2)+ 2xh ))/h`
` [ :' cos^(2) A - cos^(2)B = -sin ( A + B)sin ( A - B)]`
` = lim _( h to 0 ) [- sin ( 2x^(2)+ h^(2)+ 2xh){sinh( h + 2x)/( h + ( h + 2x))}xx ( h + 2x)]`
`= lim _( h to 0 ) { (sin h(h+2x))/( h ( h +2x)}} xx lim_( h to 0 ) [ -( h + 2x)] xx lim _( h to 0 ) sin ( 2x^(2) + h^(2) + 2xh)`
` = -1.2 sin 2x^(2) = - 2x sin ( 2x^(2))`
` d/(dx) cos^(2) x^(2) =-sin ( 2x^(2))`