The directional derivative of a real-valued function at a point in the direction of a nonzero vector is the limit

when this limit exists, where is the corresponding unit vector. If is differentiable at , then the chain rule gives

where is the gradient and is called "nabla" or "del" (Lebl 2026, pp. 40-42). Existence of the partial derivatives alone does not justify this gradient formula. All directional derivatives can exist at a point where is not differentiable (Lebl 2026, pp. 44-45).

For a differentiable function, directional differentiation is also written in the notation

where denotes a unit vector in any given direction and denotes a partial derivative.

Let be a unit vector in Cartesian coordinates, so

then

See also

Derivative, Differentiable, Gradient, Partial Derivative, Vector Derivative

Explore with Wolfram|Alpha

References

Kaplan, W. "The Directional Derivative." §2.14 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 135-138, 1991.Lebl, J. Basic Analysis II: Introduction to Real Analysis, Vol. II, version 6.3. pp. 40-45, 2026. https://www.jirka.org/ra/realanal2.pdf.Morse, P. M. and Feshbach, H. "Directional Derivatives." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 32-33, 1953.

Referenced on Wolfram|Alpha

Directional Derivative

Cite this as:

Weisstein, Eric W. "Directional Derivative." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirectionalDerivative.html

Subject classifications