Arc length equation question If you were given the following and were told to find $\dfrac{\text{d}y}{\text{d}x}$.

$y^2=2(x+4)^3$

How do you solve the problem when there is a $y^2$? When I tried solving the problem I got
$\dfrac{\text{d}y}{\text{d}x}= \dfrac{3x^2+24x+48}{y}$
I was trying to put the answer into the equation for finding the length of a curve which doesn't make sense because the equation is $\displaystyle \int_{a}^{b} \sqrt{1+\left(\dfrac{\text{d}y}{\text{d}x}\right)^2}\text{d}x$
My answer is a fraction with a $y$ variable on the bottom, but the formula at the end has a $\text{d}x$? How do you solve for $\dfrac{\text{d}y}{\text{d}x}$ so that there isn't a $y$ on the bottom fraction?

Easy enough to square root both sides:
$$y=\pm\sqrt2(x+4)^{3/2}$$
Then differentiate:
$$y'=\pm\frac32\sqrt2(x+4)^{1/2}=\frac32\sqrt{2(x+4)}$$
where the sign depends on which branch you want to compute the arc length over.
