For a simple function like $x=t^2$ you can show that the chain rule works.

$x=t^2 \to \dot x = v = 2t \to \ddot x = \dot v = a = 2$

$x=t^{1/2} \to v=2x^{1/2} \to \frac{dv}{dx} = x^{-1/2}= 1/t$

$\frac {dv}{dx} \cdot v = 1/t \cdot 2t = 2 = a$

Looking at the slopes of the graphs you can imagine that as time progresses the increasing gradient of the one on the left multiplied by the decreasing gradient of the one in the middle could produce a constant value for the graph on the right.

For a simple function like $x=t^2$ you can show that the chain rule works.

$x=t^2 \to \dot x = v = 2t \to \ddot x = \dot v = a = 2$

$t=x^{1/2} \to v=2x^{1/2} \to \frac{dv}{dx} = x^{-1/2}= 1/t$

$\frac {dv}{dx} \cdot v = 1/t \cdot 2t = 2 = a$

Looking at the slopes of the graphs you can imagine that as time progresses the increasing gradient of the one on the left multiplied by the decreasing gradient of the one in the middle could produce a constant value for the graph on the right.