Question

Find first derivative of y w.r.t. x.
(i) `y=e^(-x)` , (ii) `y=4 sin 3x` , (iii) `y=4e^(x^(2)-2x)`

Answer 1

(i) `y=e^(-x)=e^(z)` where `z=-x " "` so `(dy)/(dx)=(dy)/(dz)xx(dz)/(dx)=(e^(z))(-1)=-e^(z)=-e^(-x)`
(ii) `y=4 sin 3x=4 sinz` where `z=3x " "` so `(dy)/(dx)=(dy)/(dz)xx(dz)/(dx)=4 (cos z)(3)=12 cos 3x`
(iii) `y=4e^(x^(2)-2x)=4e^(z)` where `z=x^(2)-2x " "` so `(dy)/(dx)=(dy)/(dz)xx(dz)/(dx)=4 (e^(z))(2x-2)=(8x-x)e^(x^(2)-2x)`