is the chain rule for two inner functions, i.e...

$\displaystyle \frac{d}{dx}\ f(u(x), v(x)) = \frac{\partial f}{\partial u} \frac{du}{dx} + \frac{\partial f}{\partial v} \frac{dv}{dx}$

Or rather, in this example...

$\displaystyle \frac{d}{ds}\ f(x(s), y(s)) = \frac{\partial f}{\partial x} \frac{dx}{ds} + \frac{\partial f}{\partial y} \frac{dy}{ds}$

As with...

... the ordinary chain rule, straight continuous lines differentiate downwards (integrate up) with respect to s, and the straight dashed line similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression.

We need a similar (double) diagram to differentiate with respect to t.