निम्नलिखित फलनों को x के सापेक्ष अवकलित करें ।
(i) `e^(sin-1)x`
(ii) `e^(sin(cosx)`
(iii) `2^(sinx)`
(iv) `e^(x)+e^(x^(2))+e^(x^(3))+e^(x^(4))+e^(x^(5))`
(v) `cos^(-1)(e^(x))`

Explanation

माना कि `y=e^(sin-1)x`
तो `(dy)/(dx)=(d)/(dx)e^(sin^(-1))x=(d)/(dsin^(-1)x)esin^(-1)x*(d)/(dx)(sin^(-1)x)`
`=e^(sin^(-1))x*(1)/(sqrt(1-x^(2)))=(1)/(sqrt(1-x^(2)))e^(sin^(-1)x)`
(ii) माना कि `y=e^(sin(cosx)`
तो `(dy)/(dx)=(d)/(dx)e^(sin(cosx))=(d)/(dsin(cosx))e^(sin(cosx))*(d)/(d cosx)sin(cosx)*(d)/(dx)cosx`
`=e^(sin(cosx))*cos(cosx)*(-sinx)`
`=-sinxcos(cosx)*e^(sin)(cosx)`
माना कि `y=2^(sinx)`
तो `(dy)/(dx)=(d)/(dx)(2^(sinx))=(d)/(d(sinx))(2^(sinx))*(d)/(dx)(sinx)`
`=(2^(sinx)*log_(e)2)*cosx " " [because(d)/(dz)(a^(z))=a^(z)*log_(e)a]`
`=log_(e)2*cosx*2^(sinx)`
(iv) माना कि `y=e^(x)+e^(x^(2))+e^(x^(3))+e^(x^(4))+e^(x^(5))`
तो `(dy)/(dx)=(d)/(dx)(e^(x)+e^(x^(2))+e^(x^(3))+e^(x^(4))+e^(x^(5)))`
`=(d)/(dx)(e^(x))+(d)/(dx)(e^(x^(2)))+(d)/(dx)(e^(x^(3)))+(d)/(dx)(e^(x^(5)))+(d)/(dx)(e^(x^(5)))`
`=e^(x)+(d)/(dx^(2))(e^(x^(2)))*(d)/(dx)x^(2)+(d)/(dx^(3))(e^(x^(3)))*(d)/(dx)x^(3)+(d)/(dx^(4))(e^(x^(4)))*(d)/(dx)x^(4)+(d)/(dx^(5))(e^(x^(5)))*(d)/(dx)x^(5)`
`=e^(x)+e^(x^(2))*2x+e^(x^(3))*3x^(2)+e^(x^(4))*4x^(3)+e^(x^(5))*5x^(4)`
`=e^(x)+2x*e^(x^(2))+3x^(2)*e^(x^(3))+4x^(3)*e^(x^(4))+5x^(4)*e^(x^(5))`
(v) माना कि `y=cos^(-1)(e^(x))`
तो `(dy)/(dx)=(d)/(dx)cos^(-1)(e^(x))=(d)/(de^(x))cos^(-1)(e^(x))*(d)/(dx)(e^(x))`
`=-(1)/(sqrt(1-e^(2x)))*e^(x)=-(e^(x))/(sqrt(1-e^(2x)))`