Question

What is the derivative of `sqrt(x)/(x^3+1)` using the quotient rule?

Answer 1

We'll use the quotient rule, and, because the square root is used so often (for example: diagonal distance = hypotenuse of a right triangle), it's nice to memorize:

`d/(dx)sqrtx = 1/(2sqrtx)` (Go through it witl exponents to see that it's true.)

So:

`d/(dx)(sqrt(x)/(x^3+1)) = (1/(2sqrtx)(x^3+1) - (sqrtx) (3x^2))/((x^3+1)^2)`

Mulhtiply by `(2sqrtx)/(2sqrtx)` to get:

`d/(dx)(sqrt(x)/(x^3+1)) = ((x^3+1) - 2x (3x^2))/((2sqrtx)(x^3+1)^2) = (-5x^2+1)/((2sqrtx)(x^3+1)^2)`