Arc length equation question

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?

Answer 1

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.

Answer 2

Solve wrt $y$

$y=\pm \sqrt{2} (x+4)^{3/2}$

differentiate

$y'=\pm \frac{3\sqrt 2}{2}(x+4)^{1/2}$

Integrate $$\int_a^b \sqrt{\frac{9 (x+4)}{2}+1} \, dx=\left[\frac{4}{27} \left(\frac{9 x}{2}+19\right) \sqrt{\frac{9 x}{2}+19}\right]_a^b$$