Question

Derivative of `(2+sqrt(4-(x)^2))/x` ?

Answer 1

`(d((2+sqrt(4-x^2))/x))/dx = -(2sqrt(4-x^2)+4)/(x^2sqrt(4-x^2))`

Explanation:

Use the product rule:

`(d(g/h))/dx = ((dg)/dxh-g((dh)/dx))/h^2`

let `g = 2+sqrt(4-x^2)`, then `(dg)/dx = -x/sqrt(4-x^2)`

This implies that `h = x` and `(dh)/dx = 1`

Substituting into the rule:

`(d((2+sqrt(4-x^2))/x))/dx = (-x^2/sqrt(4-x^2)-2-sqrt(4-x^2))/x^2`

`(d((2+sqrt(4-x^2))/x))/dx = (-x^2-2sqrt(4-x^2)-(4-x^2))/(x^2sqrt(4-x^2))`

`(d((2+sqrt(4-x^2))/x))/dx = -(2sqrt(4-x^2)+4)/(x^2sqrt(4-x^2))`