1. Intuitive Idea
For a function of one variable, the derivative at a point gives the slope of the tangent line. For a function of two or more variables, there are infinitely many directions to move from a point. The directional derivative measures the rate of change in any chosen direction.
2. Formal Definition
Let \( f: \mathbb{R}^n \to \mathbb{R} \) be differentiable, and let \( \mathbf{u} \) be a unit vector (length 1) in \( \mathbb{R}^n \). The directional derivative of \( f \) at point \( \mathbf{a} \) in the direction of \( \mathbf{u} \) is:
\[ D_{\mathbf{u}} f(\mathbf{a}) = \lim_{h \to 0} \frac{f(\mathbf{a} + h\mathbf{u}) - f(\mathbf{a})}{h} \]
3. Connection to the Gradient
If \( f \) is differentiable, then: \[ D_{\mathbf{u}} f(\mathbf{a}) = \nabla f(\mathbf{a}) \cdot \mathbf{u} \] where \( \nabla f(\mathbf{a}) \) is the gradient vector of \( f \) at \( \mathbf{a} \), and “\(\cdot\)” is the dot product.
- The gradient points in the direction of steepest increase.
- The magnitude \( \|\nabla f(\mathbf{a})\| \) is the maximum possible directional derivative at \( \mathbf{a} \).
4. Steps to Compute
- Find the gradient \( \nabla f(x) \).
- Normalize the given direction vector \( \mathbf{v} \) to get \( \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \).
- Evaluate \( \nabla f \) at the given point.
- Take the dot product \( \nabla f(\mathbf{a}) \cdot \mathbf{u} \).
5. Example in 2D
Let \( f(x, y) = x^2 + y^2 \), point \( (1, 2) \), direction \( \mathbf{v} = (3, 4) \).
Normalize \( \mathbf{v} \): \[ \mathbf{u} = \left( \frac{3}{5}, \frac{4}{5} \right) \] Gradient: \[ \nabla f(x, y) = (2x, 2y) \] At \( (1, 2) \): \[ \nabla f(1, 2) = (2, 4) \] Dot product: \[ D_{\mathbf{u}} f(1, 2) = (2, 4) \cdot \left( \frac{3}{5}, \frac{4}{5} \right) = \frac{6}{5} + \frac{16}{5} = \frac{22}{5} \]
6. Example in 3D
Let \( f(x, y, z) = xyz \), point \( (1, 1, 1) \), direction \( \mathbf{v} = (1, 2, 2) \).
Normalize \( \mathbf{v} \): \[ \mathbf{u} = \frac{1}{3}(1, 2, 2) \] Gradient: \[ \nabla f(x, y, z) = (yz, xz, xy) \] At \( (1, 1, 1) \): \[ \nabla f(1, 1, 1) = (1, 1, 1) \] Dot product: \[ D_{\mathbf{u}} f(1, 1, 1) = (1, 1, 1) \cdot \frac{1}{3}(1, 2, 2) = \frac{1 + 2 + 2}{3} = \frac{5}{3} \]
7. Properties
- Linearity: \( D_{\mathbf{u}}(af + bg) = aD_{\mathbf{u}}f + bD_{\mathbf{u}}g \)
- Scaling of direction: Only the direction matters; scaling \( \mathbf{v} \) doesn’t change the result if you normalize it.
- Max/Min values: Max = \( \|\nabla f\| \) in direction of \( \nabla f \); Min = \( -\|\nabla f\| \) in opposite direction.
8. Applications
- Finding steepest ascent/descent in optimization problems.
- Measuring change of temperature, pressure, or other scalar fields in a given direction.
- Used in physics, engineering, and machine learning for sensitivity analysis.
9. Summary
The directional derivative extends the idea of a derivative to any direction in space. It’s computed as the dot product of the gradient and a unit direction vector. It tells you how fast a function changes if you move in that direction from a given point.