The Chain Rule and Directional Derivatives

Multivariable chain rule — If $z = f(x,y)$ where $x = x(t)$ and $y = y(t)$:

$$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$

More generally, if $x = x(s,t)$ and $y = y(s,t)$:

$$\frac{\partial z}{\partial s} = f_x\frac{\partial x}{\partial s} + f_y\frac{\partial y}{\partial s}$$

The gradient collects all partial derivatives into one vector:

$$\nabla f = \left(\frac{\partial f}{\partial x},\;\frac{\partial f}{\partial y}\right)$$

Directional derivative — The rate of change of $f$ in direction $\hat{u}$:

$$D_{\hat{u}}f = \nabla f \cdot \hat{u} = |\nabla f|\cos\theta$$

Key properties of the gradient

Physics link: Temperature gradient $\nabla T$ points from cold to hot. Heat flows in the opposite direction: $\vec{q} = -k\nabla T$ (Fourier's law).

Common pitfall: The directional derivative needs a unit vector: $D_{\vec{u}}f = \nabla f \cdot \hat{u}$. Feeding it an unnormalized direction silently scales the answer by that vector’s length.