# How do you calculate the derivative of y=sqrt(4x^3)?

Apr 8, 2015

You can write it as:
$y = 2 {x}^{\frac{3}{2}}$ where you used the fact that $\sqrt{x} = {x}^{\frac{1}{2}}$
So:
$y ' = 2 \cdot \frac{3}{2} {x}^{\frac{3}{2} - 1} = 3 {x}^{\frac{1}{2}} = 3 \sqrt{x}$

Apr 8, 2015

The "best" and "easiest" answer is the one given by Gio. If you want to make your life more challenging and more complicated and you'd like to spend a little more time doing things the longs way. (Or you just like using the chain rule), the you could do the following:

$y = \sqrt{4 {x}^{3}} = {\left(4 {x}^{3}\right)}^{\frac{1}{2}}$.

Therefore:

$y ' = \frac{1}{2} {\left(4 {x}^{3}\right)}^{- \frac{1}{2}} \left(12 {x}^{2}\right)$ (We used the chain rule here.)

Now we get to do a bunch of algebra to simplify the answer:

$y ' = \frac{1}{2 {\left(4 {x}^{3}\right)}^{\frac{1}{2}}} \left(12 {x}^{2}\right) = \frac{12 {x}^{2}}{2 \cdot 2 \sqrt{{x}^{3}}} = \frac{3 {x}^{2}}{\sqrt{x}} ^ 3$

$= \frac{3 {\sqrt{x}}^{4}}{\sqrt{x}} ^ 3 = 3 \sqrt{x}$

It's not the "wrong" way to find the derivative, but many people find it more complicated and more vulnerable to error.