Directional Derivatives

Rate of change along a unit vector via the gradient dot product.

The idea

The directional derivative $D_{\mathbf{u}} f$ measures the rate of change of $f(x, y)$ in an arbitrary direction. The partial derivatives report that rate along the two axes only; the directional derivative extends them to every direction.

Specify the direction as a unit vector $\mathbf{u}$. Moving a distance $h$ along $\mathbf{u}$ changes $f$ by $f(\mathbf{x} + h\mathbf{u}) - f(\mathbf{x})$, and the directional derivative is the limiting rate of change per unit of distance: $D_{\mathbf{u}} f = \lim_{h \to 0} \frac{f(\mathbf{x} + h\mathbf{u}) - f(\mathbf{x})}{h}.$ Taking $\mathbf{u} = \langle 1, 0 \rangle$ or $\langle 0, 1 \rangle$ recovers $f_{x}$ and $f_{y}$ exactly, so the two partials are two special cases among infinitely many.

We never evaluate this limit directly, because the linear approximation computes it. A step of length $h$ along $\mathbf{u} = \langle u_{1}, u_{2} \rangle$ changes $x$ by $hu_{1}$ and $y$ by $hu_{2}$, and for small $h$ each change feeds through its own partial derivative: $f(\mathbf{x} + h\mathbf{u}) \approx f(\mathbf{x}) + h u_{1}\, f_{x} + h u_{2}\, f_{y} = f(\mathbf{x}) + h\,(\nabla f \cdot \mathbf{u}),$ with an error that shrinks faster than $h$. So the difference quotient above is approximately $\nabla f \cdot \mathbf{u}$, and letting $h \to 0$ makes the approximation exact: $D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}.$

Requiring $\mathbf{u}$ to have length one makes the answer a rate per unit distance. A longer vector in the same direction scales the dot product with it, reporting a larger number for what is geometrically the same direction.

Ways to work on it

• Walkthrough. Gradient, normalize the direction, and dot to get the rate of change.
• Practice. Compute a directional derivative in a given direction.
• Hardest. Direction of steepest ascent and the maximum rate of change.

Not sure where to start? Take the ten-question placement test.