Calculator Inputs
Formula Used
This calculator estimates mixed partial derivatives numerically with repeated central differences. It works well for smooth functions and is useful when symbolic steps are unavailable.
First central difference:
∂f/∂x ≈ [f(x + h, y, z) − f(x − h, y, z)] / 2h
Mixed example, x then y:
∂²f/∂y∂x ≈ {Dxf(x, y + h, z) − Dxf(x, y − h, z)} / 2h
Symmetry check: the page also compares the chosen order with its reverse order. For smooth functions, these values often match closely.
How to Use This Calculator
- Enter a multivariable function using x, y, and optional z.
- Type the derivative order pattern, such as xy, yx, xyz, or xxy.
- Provide the evaluation point and a suitable step size.
- Set the graph span and density for the plotted surface.
- Press calculate to view the result above the form.
- Use the reverse-order value and symmetry gap to inspect order agreement.
- Download the summary as CSV or PDF when needed.
Example Data Table
| Function | Point | Order | Exact mixed partial | Notes |
|---|---|---|---|---|
| x^2*y^3 | (1, 2, 0) | xy | 24 | Differentiate by x first, then y. |
| exp(x*y) | (0, 1, 0) | yx | 1 | Smooth exponential terms usually show near-equal reversed orders. |
| x*z + y^2*z^3 | (2, 1, 1) | xz | 1 | This example uses the third variable directly. |
FAQs
1. What is a mixed partial derivative?
A mixed partial derivative differentiates a multivariable function with respect to more than one variable in sequence. Examples include xy, yx, xyz, and xxy orders.
2. Does the order always matter?
Not always. For many smooth functions with continuous lower-order partials near the point, reversing the order gives the same value. Nonsmooth or singular behavior can break that symmetry.
3. Why does this page use numeric approximation?
Numeric central differences let the calculator handle many expressions without a symbolic algebra engine. They are practical for estimating values, checking symmetry, and exploring local behavior visually.
4. What step size should I try first?
Start near 0.001 for smooth functions. If results look unstable, test slightly larger or smaller values and compare. Very tiny steps can amplify floating-point roundoff.
5. Can I use three-variable functions?
Yes. Enter x, y, and z in the expression and use patterns such as xz, yzz, or xyz. The graph is drawn over x and y while z stays fixed.
6. Which functions are supported best?
Standard arithmetic, powers, parentheses, and common functions work best. Use explicit multiplication, such as x*y, and avoid unsupported symbols or hidden multiplication like 2xy.
7. When are mixed partial derivatives piecewise functions?
They become piecewise when the original function changes formula across regions, boundaries, or absolute-value style branches. Near those transition lines, mixed orders may differ or fail to exist.
8. Why can the graph look rough near some points?
Sharp corners, discontinuities, singularities, and very small step sizes can make numerical surfaces look noisy. Increase the step slightly or inspect whether the function is smooth nearby.