Enter a function in terms of x, y, and z. This tool computes first partial derivatives, second partial derivatives, mixed partials, and (optionally) evaluates them at a point and computes a directional derivative.
Evaluation Point (optional)
Direction Vector for Directional Derivative (optional)
What this multivariable derivative calculator does
When you work with functions of several variables, the derivative becomes a richer idea. Instead of one slope, you get a slope in each coordinate direction, and together those slopes form a gradient vector. This calculator helps you quickly compute:
- First-order partial derivatives:
∂f/∂x,∂f/∂y,∂f/∂z - Second-order partial derivatives:
∂²f/∂x²,∂²f/∂y²,∂²f/∂z² - Mixed partial derivatives:
∂²f/∂x∂y,∂²f/∂x∂z,∂²f/∂y∂z - Numerical values at a point
(x, y, z), if provided - Directional derivative along a vector, if a direction is supplied
How to use it
1) Enter your function
Use standard math notation with operators and functions like +, -, *, /, ^, sin(), cos(), exp(), and ln().
2) (Optional) Enter a point
If you enter values for variables used in your function, the tool evaluates the function and its derivatives numerically at that point.
3) (Optional) Enter a direction vector
If you provide (vx, vy, vz), the calculator computes the directional derivative using the unit vector in that direction.
Why these derivatives matter
Multivariable derivatives appear everywhere: optimization in machine learning, sensitivity analysis in engineering, local linear approximations in physics, and error propagation in applied statistics. In practical terms:
- Gradient points in the direction of steepest increase.
- Second derivatives describe curvature and are useful for classifying critical points.
- Mixed partials capture how variables interact.
- Directional derivatives measure change along a chosen path.
Worked example
Suppose:
f(x,y,z) = x^2 y + sin(xy) + z^3
At point (1,2,0), the calculator can return exact symbolic expressions and their numeric values. If you also choose direction (1,-1,2), it computes the rate of change of f in that direction.
Tips for best results
- Use explicit multiplication: write
x*y(notxy). - Use parentheses for clarity:
sin(x*y). - If you only need symbolic derivatives, leave point values blank.
- For directional derivatives, avoid the zero vector.
Frequently asked questions
Does this handle functions with only x and y?
Yes. You can input functions that use any subset of x, y, z. Unused-variable derivatives are automatically zero.
Can I use constants like pi and e?
Yes. You can type expressions like sin(pi*x) or e^(x+y).
Are results exact or approximate?
Symbolic derivatives are exact expressions. Numerical evaluations are approximate decimal values based on your point input.