oh ! so is my answer incorrect ?
For that last example, you overlooked squaring the entire denominator.
What I meant earlier is that you are comfortable with the situation
where the "function nesting" is one level deep, when applying the chain rule.
$\displaystyle e^{6x}$ is one level beyond $\displaystyle e^x$
q=6x
$\displaystyle \frac{d}{dx}e^{6x}=\frac{d}{dx}e^q=\frac{dq}{dx} \frac{d}{dq}e^q$
However...
$\displaystyle v=\left(1+3x^2\right)^{\frac{1}{2}}$
has an extra stage.
Beginning with x, we have
$\displaystyle w=1+3x^2$
but then we take the square root of this
$\displaystyle v=w^{\frac{1}{2}}$
so the "nesting" is 2 levels deep, whereas it was only 1 level deep for $\displaystyle e^{6x}$
Hence
$\displaystyle \frac{d}{dx}\left(1+3x^2\right)^{\frac{1}{2}}= \frac{d}{dx}w^{\frac{1}{2}}=\frac{dw}{dx}\frac{d}{ dw}w^{\frac{1}{2}}$
$\displaystyle =\frac{dw}{dx}\left[\frac{1}{2}w^{-\frac{1}{2}}\right]$
How is that ?
Ok I will do squaring and see if I can simplify in any way. However I take it rest of is ok. Is that correct? Also I know why I made mistake in "v" because I wasn't careful but since you pointed it out I think I will be extra carful in writing my steps out. Just away from the computer for next hour so when get back will send you my correction. I must say you have given me that confidence I thought I'll never have . Sincere thank you