Question

What is the derivative of `arctan(x)/(1+x^2)`?

Answer 1

`d/dx(arctan(x)/(1+x^2))=(1-2xarctan(x))/(1+x^2)^2`

Explanation:

The quotient rule states that the derivative of a function `f(x)=g(x)/(h(x))` is:

`f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x))^2`

Here we see that:

• `g(x)=arctan(x)" "=>" "g'(x)=1/(1+x^2)`
• `h(x)=1+x^2" "=>" "h'(x)=2x`

Then:

`d/dx(arctan(x)/(1+x^2))=((1/(1+x^2))(1+x^2)-arctan(x)(2x))/(1+x^2)^2`

`color(white)(d/dx(arctan(x)/(1+x^2)))=(1-2xarctan(x))/(1+x^2)^2`