Question

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

Answer 1

You can write it as:
`y=2x^(3/2)` where you used the fact that `sqrt(x)=x^(1/2)`
So:
`y'=2*3/2x^(3/2-1)=3x^(1/2)=3sqrt(x)`

Answer 2

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(4x^3) = (4x^3)^(1/2)`.

Therefore:

`y' = 1/2(4x^3)^(-1/2)(12x^2)` (We used the chain rule here.)

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

`y' = 1/(2(4x^3)^(1/2))(12x^2) = (12x^2) /( 2*2sqrt(x^3))=(3x^2)/sqrtx^3`

`= (3sqrtx^4)/sqrtx^3 = 3sqrtx`

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