x के सापेक्ष निम्नलिखित फलनों को अवकलित करें ।
(i) `cosx^(3)*sin^(2)(x^(5))`
(ii) `sqrt(x sinx)`
(iii) `(sin(ax+b))/(cos(cx+d))`
(iv) `sinsqrt(x)+cos^(2)sqrt(x)`

Explanation

(i) माना कि `y=cosx^(3)*sin^(2)(x^(5))` तो `(dy)/(dx)=cosx^(3)*(d)/(dx){sin^(2)(x^(5))}+sin^(2)(x^(5))*(d)/(dx)(cosx^(3))`
`=cosx^(3)*(d)/(dsin(x^(5)))sin^(2)(x^(5))*(d)/(dx^(5))sin(x^(5))*(d)/(dx)x^(5)+sin^(2)(x^(5))*(d)/(dx^(3))cosx^(3)*(d)/(dx)x^(3)`
`=cosx^(3)*2sin(x^(5))*cos(x^(5))5x^(4)+sin^(2)(x^(5))*(-sinx^(3))*3x^(2)`
`=5x^(4)cosx^(3)*sin(2x^(5))-3x^(2)sinx^(3)sin^(2)(x^(5))`
(ii) माना कि `y=sqrt(x sinx)`
तो `(dy)/(dx)=(d)/(dx)sqrt(xsinx)=(d)/(d(xsinx))sqrt(x sinx)*(d)/(dx)(x sinx)`
`=(1)/(2sqrt(xsinx))*[x*(d)/(dx)(sinx)+(d)/(dx)(x)*sinx]`
`=(1)/(2sqrt(xsinx))[x cosx+sinx]`
(iii) माना कि `y=(sin(ax+b))/(cos(cx+d))`
तो `(dy)/(dx)=(cos(cx+d)*(d)/(dx)sin(ax+b)-sin(ax+b)(d)/(dx)cos(cx+d))/(cos^(2)(cx+d))`
`=(1)/(cos^(2)(cx+d))*[cos(cx+d)*(d)/(d(ax+b))sin(ax+b)*(d)/(dx)(ax+b)`
`-sin(ax+b)*(d)/(d(cx+d))cos(cx+d)*(d)/(dx)(cx+d)`
`=sec^(2)(cx+d)*[cos(cx+d)cos(ax+b)*a`
`-sin(ax+b)*{-sin(cx+d)}c]`
`=asec(cx+d)cos(ax+b)+csin(ax+b)sin(cx+d)sec^(2)(cx+d)`
(iv) माना कि `y=sinsqrt(x)+cos^(2)sqrt(x)`
तो `(dy)/(dx)=(d)/(dx)sinsqrt(x)+(d)/(dx)cos^(2)sqrt(x)`
`=(d)/(dsqrt(x))sinsqrt(x)*(d)/(dx)sqrt(x)+(d)/(d cossqrt(x))cos^(2)sqrt(x)*(d)/(dsqrt(x))cossqrt(x)*(d)/(dx)sqrt(x)`
`=cossqrt(x)*(1)/(2sqrt(x))+2cossqrt(x)(-sinsqrt(x))*(1)/(2sqrt(x))`
`=(1)/(2sqrt(x))(cossqrt(x)-sin2sqrt(x)) " " [because2sinthetacostheta=sin2 theta]`