Question

What is the derivative of `(xe^-x)/(x^3+x)`?

Answer 1

Let's use product rule, here, shall we?

`f(x)=(xe^-x)(x^3+x)^-1`

We'll also need product rule to find the first term's derivative and chain rule for the second one's.

Rules:

• Product rule: be `y=f(x)g(x)`, then `y'=f'(x)g(x)+f(x)g'(x)`
• Chain rule: `(dy)/(dx)=(dy)/(du)(du)/(dx)`

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

#(dy)/(dx)=e^-x(-x^4-x^2-2x^3)

`(dy)/(dx)=((1-x)(x^3+x))/e^x-(3x^3+x)/e^x=-(x^4+2x^3+x^2)/e^x=(x^2(x^2+2x+1))/e^x=color(green)((x^2(x+1)^2)/e^x)`