The Chain Rule and Directional Derivatives

Multivariable Chain Rule

When a function depends on variables that themselves depend on time or parameters, we use the multivariable chain rule. If $z = f(x,y)$ where $x = x(t)$ and $y = y(t)$:

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

For functions of multiple intermediate variables like $x(s,t)$ and $y(s,t)$:

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

The gradient collects all partial derivatives into a single vector: $\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$.

Directional Derivatives & Gradient

The directional derivative measures the rate of change of $f$ in the direction of a unit vector $\hat{u}$:

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

Common pitfall: The directional derivative requires a unit vector ($\hat{u}$). Feeding it an unnormalized vector silently scales your answer.