Start with the time rate of change of the kinetic energy:

K= 1/2mv^2

dK/dt = d/t(1/2mv^2) = m*(dv/dt)*v

F = ma = m (dv/dt) Newtons Second Law ...

dK/dt = F(dx/dt) since v=dx/dt

dK = F*dx

Now integrate from initial position x_0 to final position x_f.

integral of dK from K_0 to K_f = integral of F dx from x_0 to x_f

or

K_f - K_i = integral of F dx from x_0 to x_f

but the rhs is just the definition of work, so we get the Work-Energy Theorem:

K_f - K_i = W

from the integral form of Newtons Second Law per above.