So I completely understand how $a = v(dv/dx)$. But how does differentiating $\frac12 v^2$ with respect to $x$ get you to $v(dv/dx)$. Can someone put all steps in please?
Why can you say $d(\frac12 v^2)/dx = \frac12 \times 2v \times dv/dx$ (where does $2v$ and $dv/dx$ come from? Is it actually written properly as $\frac12 d(v^2)/dx$? Then I'm assuming you take the differential out i.e. $2v$ to get the $2v$?
On
If you're in a mechanics course you should already have at least Calculus I under your belt. This is just a basic derivative,
$\frac{d}{dx}( x^2 ) = 2x $.
Now if you are taking the derivative of something that depends on $x$, like $v=v(x)$ in your case,
$\frac{d}{dx} ( f(x)^2 ) = 2f(x)\frac{df(x)}{dx}$,
from the chain rule.
Well, if you know that
$$a=v \frac{dv}{dx},$$
you can prove it like this:
$$\frac{1}{2} \frac{dv^2}{dx}=\frac{1}{2} \frac{dv^2}{dv}\frac{dv}{dx}=\frac{2v}{2} \frac{dv}{dx}=v\frac{dv}{dx}=a.$$