Question

How do you differentiate `(xsinx)/(x^2+1)`?

Answer 1

We'll be using the product rule with the quotient rule:

Let's think of your function as:

`(f(x)*g(x))/(h(x))`

Where:
`f(x)=x`
`g(x)=sin(x)`
`h(x)=x^2+1`

Thus, the derivative is:

`[d/dx(f(x)*g(x))*h(x)-(f(x)*g(x))*d/dx(h(x))]/(h(x)^2)`

Plug in our functions and find the derivatives:

`[d/dx(xsinx)*(x^2+1)-(xsinx)*d/dx(x^2+1)]/((x^2+1)^2)`

`=[(sinx+xcosx)(x^2+1)-(2x^2sinx)]/(x^2+1)^2`