A directional derivative measures how fast a multivariable function changes as you move away from a point in a chosen direction. For a surface z = f(x, y), it is the slope you feel if you stand on the surface and walk in a particular direction in the xy-plane. This matters because real quantities such as temperature, height, pressure, and electric potential often change differently depending on direction.
Directional derivatives connect local geometry, rates of change, and optimization in one useful idea.
The most efficient way to compute a directional derivative is to use the gradient vector and a unit direction vector. If u is a unit vector, then D_u f(a, b) = grad f(a, b) · u, so the directional derivative is a dot product. The gradient points in the direction of greatest increase, its opposite gives greatest decrease, and any direction perpendicular to it gives zero instantaneous change.
This makes directional derivatives useful for contour maps, tangent planes, steepest ascent paths, and local linear approximation.
Key Facts
- Directional derivative definition: D_u f(a, b) = lim h->0 [f(a + h u1, b + h u2) - f(a, b)] / h.
- Gradient in two variables: grad f(x, y) = <f_x(x, y), f_y(x, y)>.
- Computing formula: D_u f(a, b) = grad f(a, b) · u, where u is a unit vector.
- Unit vector requirement: if v = <p, q>, then u = v / |v| = <p, q> / sqrt(p^2 + q^2).
- Maximum directional derivative: max D_u f(a, b) = |grad f(a, b)|, in the direction of grad f(a, b).
- Minimum directional derivative: min D_u f(a, b) = -|grad f(a, b)|, in the direction of -grad f(a, b).
Vocabulary
- Directional derivative
- The instantaneous rate of change of a function at a point as you move in a specified direction.
- Gradient
- A vector made from the partial derivatives of a function that points in the direction of steepest increase.
- Unit vector
- A vector with length 1 that gives direction without changing the scale of the rate of change.
- Tangent plane
- The plane that best approximates a smooth surface near a given point.
- Level curve
- A curve in the input plane along which a function has a constant value.
Common Mistakes to Avoid
- Using a non-unit direction vector, which makes the directional derivative too large or too small because the vector length scales the dot product.
- Confusing the gradient with the directional derivative, which is wrong because the gradient is a vector while the directional derivative is a scalar rate.
- Forgetting to evaluate the gradient at the given point, which gives a formula for many points instead of the requested local rate of change.
- Assuming the steepest direction is always along the x-axis or y-axis, which is wrong because the steepest direction depends on both partial derivatives together.
Practice Questions
- 1 For f(x, y) = x^2 + 3y^2, find D_u f(1, 2) in the direction u = <3/5, 4/5>.
- 2 For f(x, y) = e^x y, find the directional derivative at (0, 2) in the direction of v = <1, -1>.
- 3 At a point, grad f = <6, 8>. Explain which direction gives the greatest increase, which gives the greatest decrease, and which directions give zero instantaneous change.