Deriving work-energy theorem from Newton's second law

[negation]

Homework Statement

I'm trying to bridge F =ma to m/2(dv²/dx). It was shown in the course book I have but there's a huge disconnection in the steps.

The Attempt at a Solution

F =ma = m.(dv/dt) = m(dv/dx . dx/dt) = mv(dv/dx). Where do I take it from here?

 

[rude man]

Formally differentiate d/dx(v^2). What do you get?

 

[dauto]

Homework Statement

I'm trying to bridge F =ma to m/2(dv²/dx). It was shown in the course book I have but there's a huge disconnection in the steps.

The Attempt at a Solution

F =ma = m.(dv/dt) = m(dv/dx . dx/dt) = mv(dv/dx). Where do I take it from here?

Multiply both sides of your equation by dx to get

F dx = mv dv

Integrate both side, what do you get?

 

[negation]

Multiply both sides of your equation by dx to get

F dx = mv dv

Integrate both side, what do you get?

Both sides?
I could arrive at the conclusion if I were to integrate only the left argument.

∫F(x).dx = ∫ma.dx = ∫m(dv/dt).dx = ∫m(dv/dx . dx/dt) .dx = ∫m(dv/dx . v).dx = ∫m*dv.v = ∫mv.dv = m∫dv . ∫v.dv = m (0.5v²) = 0.5mv² + C

 

[negation]

Formally differentiate d/dx(v^2). What do you get?

derivative of v² = 2v(dv/dx)