For a function $$f(x, y)$$ and a curve $$\gamma (t) = (x(t), y(t))$$ the chain rule gives
$\dfrac{df(\gamma (t))}{dt} = \left. \dfrac{\partial f}{\partial x} \right\vert_{\gamma (t)} x'(t) + \left. \dfrac{\partial f}{\partial y} \right \vert_{\gamma (t)} y'(t) = \nabla f(\gamma (t)) \cdot y'(t) \text{ dot product of vectors.}$
Here $$\nabla f$$ is the gradient of $$f$$ defined in the next section.