Question

In Exercise, find the derivative of the function.
y = 3ex - xex

Answer 1

the question is incomplete, the complete question is "find the derivative of the function
`y=3e^(x)-xe^(x)`"

answer:
`(dy)/(dx)=(2-x)e^(x)`.

Explanation:

From the equation,
`y=3e^(x)-xe^(x)`, we approach the question using the differentiation of a product and differentiation of a sum simultaneously,

the differentiation of a sum is express as

f(x)=u(x)+v(x)+.....w(x) then

`(df(x))/(dx)=(du(x))/(dx)+(dv(x))/(dx)+...(dw(x))/(dx).\\`

For the differentiation of a product we have

f(x)=u(x)v(x), then

`(df(x))/(dx)=(dv(x))/(dx)u(x)+(du(x))/(dx)v(x)`

hence if we go by the formula we arrive at

for
`y=3e^(x)`

let u(x)=3 hence du/dx=0 and

`v(x)=e^(x)` and
`(dv(x))/(dx)=e^(x)`

hence
`(dy)/(dx)=3e^(x)+0e^(x)\\(dy)/(dx)=3e^(x)---equation 1\\`

Also for
`y=-xe^(x)\\(dy)/(dx)=-e^(x)-xe^(x)---equation 2`.

if we add equation 1 and equation 2 we arrive at

`(dy)/(dx)=3e^(x)-e^(x)-xe^(x)\\(dy)/(dx)=(3-1-x)e^(x)\\(dy)/(dx)=(2-x)e^(x)`.