Given:

The given relation is a(t)=v(t)dvds , where

  a(t)= acceleration.

  v(t)= velocity

  s(t)= displacement

The given derivatives are dvdt and dvds .

Calculation:

Acceleration is the rate of change of velocity with respect to time.

  a(t)=dvdt

For increasing velocity, the sign of dvdt will be positive and for the decreasing velocity, the sign of dvdt will be negative.

Now,

  a(t)=dvdt

Divide both sides by v .

  ⇒1v⋅a(t)=dvdt⋅1v⇒1v⋅a(t)=dvdt⋅dtds

Apply chain rule.

  dfda⋅dadx=df(a)dx

  ⇒1v⋅a(t)=dvds⇒a(t)=vdvds⇒a(t)=v(t)dvds

  dvds is the rate of change of the velocity with respect to displacement.

Whenthe displacement increases, the velocity is increasing, when the displacement decreases, the velocity is decreasing.

Hence a(t)=v(t)dvds proved.