Question

How do you differentiate `f(x) =x^2/(e^(3-x)+2)` using the quotient rule?

Answer 1

`color(blue)(f' (x)=(x^2*e^(3-x)+2x*e^(3-x)+4x)/(e^(3-x)+2)^2)`

Explanation:

Start from the given `f(x)=x^2/(e^(3-x)+2)`

Use the formula `d/dx(u/v)=(v*d/dx(u)-u*d/dx(v))/v^2`

Let `u=x^2` and `v=e^(3-x)+2`

`f' (x)=d/dx(x^2/(e^(3-x)+2))`

`f' (x)=((e^(3-x)+2)*d/dx(x^2)-x^2*d/dx(e^(3-x)+2))/(e^(3-x)+2)^2`

`f' (x)=((e^(3-x)+2)*2x-x^2(e^(3-x)(0-1)+0))/(e^(3-x)+2)^2`

`f' (x)=(2xe^(3-x)+4x+x^2e^(3-x))/(e^(3-x)+2)^2`

then rearranging the numerator

`color(red)(f' (x)=(x^2e^(3-x)+2xe^(3-x)+4x)/(e^(3-x)+2)^2)`

God bless....I hope the explanation is useful.