Working on a bit of mechanics at the moment. I'm trying to arrive at a textbook formula but am taking a slightly different approach, however, I don't arrive at the right answer and was hoping one of you guys could find where I go wrong.
(v^2)(d/dx) = (v^2)(dt/dx)(d/dt) : Using chain rule = (v^2)(1/v)(d/dt) = (v)(d/dt) = a
Thanks in advance for any help.
I think this is what you are looking for: $$\frac{{d\left( {{v^2}} \right)}}{{dx}} = \frac{{d\left( {{v^2}} \right)}}{{dv}}\frac{{dv}}{{dx}} = 2v\frac{{dv}}{{dx}}$$ $$v = \frac{{dx}}{{dt}}$$ $$\boxed{\frac{{d\left( {{v^2}} \right)}}{{dx}} = 2v\frac{{dv}}{{dx}} = 2\frac{{dx}}{{dt}}\frac{{dv}}{{dx}} = 2\frac{{dv}}{{dt}} = 2a}$$
If the first step confuses you, let $u = v^2$: $$\frac{{du}}{{dv}} = \frac{d}{{dv}}\left( {{v^2}} \right) = 2v$$\ $$\frac{{du}}{{dx}} = \frac{{du}}{{dv}}\frac{{dv}}{{dx}} = 2v\frac{{dv}}{{dx}}$$