Calculator form
This page uses a single-column content flow. The input grid becomes three columns on large screens, two on medium screens, and one on mobile devices.
Formula used
If z = F(u, v), with u = u(x, y) and v = v(x, y), then:
∂z/∂x = (∂F/∂u)(∂u/∂x) + (∂F/∂v)(∂v/∂x)
∂z/∂y = (∂F/∂u)(∂u/∂y) + (∂F/∂v)(∂v/∂y)
This calculator models both inner functions as quadratic surfaces in x and y. It then evaluates the outer function derivative with respect to u and v, before combining both dependency paths.
The sensitivity shares compare the absolute size of each path contribution. That helps you see whether the change comes mostly from the u branch or the v branch.
How to use this calculator
- Choose the outer function model that connects u and v to z.
- Set integer powers n and m when your chosen model uses exponents.
- Enter coefficients for u(x,y) and v(x,y).
- Provide the evaluation point (x₀, y₀).
- Set the graph range and fixed y value for visualization.
- Press the calculate button to show z, partial derivatives, step breakdown, and the chart.
- Use the CSV and PDF buttons to export the computed summary.
Example data table
These examples show valid setups you can test quickly. Values are rounded.
| Outer model | u(x,y) | v(x,y) | Point | ∂z/∂x | ∂z/∂y |
|---|---|---|---|---|---|
| z = u²v² | x² + 0.5y² - 0.2xy + 1 | 0.8x² + 1.1y² + 0.3xy + 0.5 | (1, 2) | 816.48 | 1281.8736 |
| z = sin(u) + cos(v) | 1.2x² + 0.4y² + 0.6xy + 0.2 | 0.5x² + 0.9y² - 0.3xy + 1.1 | (0.5, 1.2) | 0.157538 | -1.273386 |
| z = ln(1 + u² + v²) | 0.9x² + 0.2y² + 0.7xy + 1 | 0.6x² + y² - 0.4xy + 0.3 | (1.5, -0.5) | 1.684019 | -0.219161 |
Frequently asked questions
1. What does this calculator solve?
It computes multivariable chain rule derivatives for z = F(u,v), where both inner functions depend on x and y. You get z, partial derivatives, sensitivity paths, a summary table, and a graph.
2. Why are there two inner functions?
Many composite models depend on several intermediate variables. The chain rule adds each dependency path, so separate u and v functions help visualize how every branch contributes to the final derivative.
3. Why are powers limited to integers?
Integer powers keep symbolic differentiation stable for negative inner values and avoid branch issues that appear with fractional exponents. That makes the calculator more predictable for learning and testing.
4. What do the sensitivity percentages mean?
They compare absolute contribution sizes from the u path and v path. A larger share means that branch has more influence on the final derivative at the chosen evaluation point.
5. What happens in the quotient model?
The quotient model uses z = u / (v + s). The shift s moves the denominator away from zero. If v + s becomes zero, the calculator blocks the result and shows a warning.
6. Why graph against x with fixed y?
That view isolates one input direction while preserving multivariable structure. It helps you compare the function value and both partial derivatives across a controlled slice of the surface.
7. Can I use this for teaching?
Yes. The step-by-step section is designed for class demonstrations, homework checking, and self-study. It shows inner derivatives, outer derivatives, and the exact chain rule combination.
8. Does this calculator replace symbolic algebra software?
No. It focuses on a structured family of multivariable chain rule problems. That makes it fast, readable, and practical, but less general than a full symbolic computation system.