माना कि `y=cos(sinx)`
अब `(dy)/(dx)=(d)/(dx){cos(sinx)}=(d)/(dsinx){cos(sinx)}*(d)/(dx)(sinx)`
`=-sin(sinx)*cosx=-cosxsin(sinx)`
(ii) माना कि `y=cossqrt()x`
अब `(dy)/(dx)=(d)/(x)(cossqrt(x))=(dcossqrt(x))/(dsqrt(x))*(dsqrt(x))/(dx)`
`=-sinsqrt(x)*(1)/(2sqrt(x))=-(1)/(2sqrt(x))sinsqrt(x)`
(iii) माना कि `y=tan=tan(2x+3)`
अब `(dy)/(dx)=(d)/(dx)tan(2x+3)=(d tan(2x+3))/(d(2x+3))*(d)/(dx)(2x+3)`
`=sec^(2)(2x+3)*2=2sec^(2)(2x+3)`
(iv) माना कि `y=sin(x^(2)+5)`
तो `(dy)/(dx)=(d)/(dx)sin(x^(2)+5)=(d)/(d(x^(2)+5))sin(x^(2)+5)*(d)/(dx)(x^(2)+5)`
`=cos(x^(2)+5)*[(d)/(dx)(x^(2))+(d)/(dx)(5)]`
`=cos(x^(2)+5)*(2x+0)=2xcos(x^(2)+5)`
(v) माना कि `y=sec(tansqrt(x))`
तो `(dy)/(dx)=(d)/(dx)(sec(tansqrt(x)))`
`=(d)/(d tansqrt(x)){(sec(tansqrt(x)))}*(d)/(dsqrt(x))(tansqrt(x))*(d)/(dx)sqrt(x)`
`=sec(tansqrt(x))tan(tansqrt(x))*sec^(2)sqrt(x)*(1)/(2sqrt(x))`
`=(sec^(2)sqrt(x))/(2sqrt(x))sec(tansqrt(x))*tan(tansqrt(x))`
(vi) माना कि `y=sin(cosx^(2))`
तो `(dy)/(dx)=(d)/(dx)sin(cosx^(2))=(d)/(dcosx^(2))sin(cosx^(2))*(d)/(dx^(2))cosx^(2)*(d)/(dx)x^(2)`
`=cos(cosx^(2))*(-sinx^(2))*2x`
`=-2xsinx^(2)cos(cosx^(2))`
(vii) माना कि `y=2sqrt(cot(x^(2)))`
तो `(dy)/(dx)=(d)/(dx)(2sqrt(cotx^(2)))=2(d)/(dx)sqrt(cotx^(2))`
`=2(d)/( d cotx^(2))sqrt(cotx^(2))*(d)/(dx^(2))cotx^(2)*(d)/(dx)x^(2)`
`=2*(1)/(2sqrt(cotx^(2)))*(-cosec^(2)x^(2))*2x`
`=-(2x)/(sqrt(cot)x^(2))cosec^(2)x^(2)`
