Answer
See below for proof that the acceleration of the particle is $f(x)f'(x)$.
Work Step by Step
Velocity of the particle: $$v=\frac{dx}{dt}=f(x)$$
Acceleration of the particle: $$a=\frac{dv}{dt}$$
Applying the Chain Rule, we can say that $$a=\frac{dv}{dx}\frac{dx}{dt}$$
Recall above that $dx/dt=f(x)$: $$a=\frac{dv}{dx}\times f(x)$$
For $dv/dx$: $$\frac{dv}{dx}=\frac{d}{x}f(x)=f'(x)$$
Therefore, $$a=f(x)f'(x)$$
