Gradient Calculator: Gradient, Laplacian and Directional Derivative of f(x, y, z) at a Point
Partial derivatives, gradient and Laplacian at a point.
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How this calculator works
Type the function as an expression in x, y and z.
Enter the point and the direction.
The calculator gives the partial derivatives, the gradient, the Laplacian and the directional derivative.
Formulas used
- \nabla f = \left(\frac{\partial f}{\partial x},\ \frac{\partial f}{\partial y},\ \frac{\partial f}{\partial z}\right)
- \nabla^2f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} + \frac{\partial^2f}{\partial z^2}
- D_{\mathbf u}f = \nabla f\cdot\mathbf u
Worked example
With the default numbers in the calculator above:
- Value of the function at the point: 2
- Partial derivative with respect to x: 4
- Partial derivative with respect to y: 1
- Partial derivative with respect to z: 1
- Size of the gradient: 4.24264
- Laplacian: 4
- Directional derivative along your direction: 3.4641
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What does the gradient tell me?
It points in the direction in which the function increases fastest, and its size is the rate of increase in that direction.
What is the Laplacian?
The sum of the second partial derivatives. It appears in heat flow, wave and electrostatic equations.
What is a directional derivative?
The rate of change of the function along a chosen direction. It is the gradient dotted with a unit vector in that direction.
Related calculators
Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any multivariable calculus text (the gradient and directional derivative).
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