Gradient Vector Calculator
The gradient of a scalar function bundles its three partial derivatives into a single vector that points in the direction of steepest increase. Its length is the maximum rate of change at the point, and dividing by that length gives the unit direction of fastest ascent. This calculator takes the three partial derivatives, evaluated at your point of interest, and returns the gradient vector, its magnitude, and the unit gradient direction. The gradient underpins optimization, gradient descent, level surfaces, and directional derivatives.
Gradient formula
grad f = ( df/dx, df/dy, df/dz )
|grad f| = sqrt( (df/dx)^2 + (df/dy)^2 + (df/dz)^2 )
unit = grad f / |grad f|
points toward steepest increase
negative gradient points toward steepest decrease
The gradient stacks the three partial derivatives into a vector. Its magnitude is the steepest slope, and normalizing it gives the direction of fastest increase.
Notes on the gradient
- The three inputs are partial derivatives already evaluated at your chosen point.
- The gradient points in the direction of steepest increase of the function.
- Its magnitude is the maximum rate of change at the point.
- The gradient is perpendicular to the level surface through the point.
- If all derivatives are zero the gradient is the zero vector and the direction is undefined.
Gradient: frequently asked questions
What is the gradient of a scalar function?
The gradient is the vector of all first-order partial derivatives of a scalar function. It points in the direction of steepest increase of the function, and its magnitude equals the maximum rate of increase at that point.
What is the gradient formula?
For a function f of x, y, and z, the gradient is the vector (df/dx, df/dy, df/dz). Each component is the partial derivative of f with respect to one variable, holding the others fixed.
What does the gradient magnitude represent?
The magnitude of the gradient is the steepest slope of the function at the point, the greatest rate of change in any direction. It equals the square root of the sum of the squared partial derivatives.
How is the gradient related to direction of steepest ascent?
The gradient points in the direction of fastest increase. The unit gradient, the gradient divided by its magnitude, gives that direction as a unit vector. The negative gradient points toward fastest decrease.
Why enter partial derivatives rather than a function?
The gradient is defined exactly by the partial derivatives evaluated at your point. Supplying those three values keeps the result deterministic and exact without parsing arbitrary symbolic functions.
Official sources
- NIST Digital Library of Mathematical Functions: Vectors and vector-valued functions.
- NIST Digital Library of Mathematical Functions: Calculus of two or more variables.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.