Derivative of a function of a function - .

[christian0710]

Hi at 1 Hour and 9 minutes this professor makes a derivation which i do not understand

He is lecturing on Newtonian mechanics and states that if

dv/dt = a (acceleration)

Then

v*dv/dt = a*v

And then he says that this is the same as

d(v^2/2)/dt

But I just can't undrestand how he did that last part? I know how to apply the chain rule to the derivative of a funnction of a function but i can't see how this applies in this example. The form he writes it on is f(x)*d(f(x)/dx (the same as v*dv/dt ) and the chain rule is on the form df(g(x))/dx = df(g)/dg * d(g(x))/dx? did he write it in a wrong mathematical form? What rule is he using?

 

[jedishrfu]

If you work it backward you get what he started with right?

ie d( x^2) / dx = 2x and so 1/2 d( x^2 ) = x

let u = v^2

then du/dt = 2v dv/dt right?

 

[BvU]

All he says is ${d\over dt} \left (v^2/2\right ) = v {dv\over dt}$ right ?

the rule he is using is that d/dg (g²/2) = g !


if f = g²/2 and g = v your chain rule also says df/dg = g and df/dt = g dg/dt , in other words d/dt(v² /2) = v dv/dt