Calculator
Example Data Table
| Example | Expression | x | y | u | v | dx/dt | dy/dt | du/dt | dv/dt |
|---|---|---|---|---|---|---|---|---|---|
| 1 | x^2*y + sin(u) + exp(v) | 2 | 3 | 0.5 | 0.2 | 1.2 | -0.8 | 0.4 | 0.1 |
| 2 | sqrt(x+y) + u^3 - log(v+2) | 1.5 | 2.5 | 1 | 0.6 | 0.7 | 1.1 | -0.3 | 0.05 |
| 3 | sin(x*y) + cos(u*v) + x/v | 1.2 | 0.9 | 0.4 | 2.2 | 0.2 | -0.4 | 0.8 | -0.1 |
Formula Used
Total derivative with respect to t:
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) + (∂z/∂u)(du/dt) + (∂z/∂v)(dv/dt)
Numerical partial derivative:
∂z/∂x ≈ [f(x+h, y, u, v) − f(x−h, y, u, v)] / (2h)
This calculator uses central differences to estimate partial derivatives, then applies the chain rule to combine those partial changes with the entered rates.
How to Use This Calculator
- Enter the multivariable expression for z using x, y, u, and v.
- Provide the current values of x, y, u, and v.
- Enter the rates dx/dt, dy/dt, du/dt, and dv/dt.
- Choose a small numerical step for derivative estimation.
- Set graph sample points and direction span for the Plotly charts.
- Click the calculate button to show the result above the form.
- Review partial derivatives, variable contributions, and total derivative.
- Download the result table as CSV or PDF when needed.
Frequently Asked Questions
1. What does a total derivative measure?
It measures how a function changes when all input variables change together with respect to one independent variable, usually time.
2. Why are partial derivatives shown separately?
Separate partials reveal how sensitive the function is to each variable before combining those sensitivities with the entered rates.
3. Why does this tool ask for dx/dt and other rates?
The chain rule needs both local sensitivity and variable motion. Rates tell the calculator how fast each variable changes with time.
4. Can I use trigonometric and logarithmic functions?
Yes. The calculator supports sin, cos, tan, asin, acos, atan, sqrt, ln, log, exp, and abs, plus constants pi and e.
5. Why is the derivative numerical instead of symbolic?
Numerical differentiation keeps the calculator flexible for many valid expressions and avoids building a full symbolic algebra engine.
6. How should I choose the step size?
Use a small positive value like 0.0001. Extremely large steps reduce accuracy, while extremely tiny steps may amplify rounding errors.
7. What does the contribution chart show?
It shows the signed amount each variable adds to dz/dt. Positive bars increase total change. Negative bars decrease it.
8. When is this calculator useful?
It helps in physics, optimization, economics, engineering, and multivariable calculus whenever several quantities vary simultaneously.