How to Calculate Partial Derivatives
In multivariable calculus, a partial derivative represents the rate of change of a function with respect to one specific variable while holding all other variables constant. This calculator evaluates the exact numerical derivative of your function at a designated point (x, y) using the central difference quotient.
- ∂f/∂x = Slope of the function moving along the x-axis.
- ∂f/∂y = Slope of the function moving along the y-axis.
- h = An infinitesimally small step size (this calculator uses h = 10-5).
When solving partial derivatives by hand (symbolically), you apply standard differentiation rules to the target variable and treat the secondary variable exactly as if it were a plain number (a constant).
| Variable | Rule Application | Result |
|---|---|---|
| w.r.t x (∂f/∂x) | Treat y as a constant. The derivative of x² is 2x. The derivative of 3y is 0. | 2xy |
| w.r.t y (∂f/∂y) | Treat x as a constant. The derivative of y is 1. The derivative of 3y is 3. | x² + 3 |
Frequently Asked Questions
What is the difference between a partial and total derivative?
A standard (total) derivative is used for functions of a single variable, representing the total rate of change. A partial derivative is used for functions of multiple variables (like 3D surfaces), measuring the rate of change in one specific direction (e.g., exclusively along the x-axis) while ignoring changes in other directions.
What does the ∂ symbol mean?
The "∂" (del or partial) symbol differentiates the notation from a standard "d". While df/dx implies a total derivative, ∂f/∂x explicitly signals to the reader that the function has multiple variables and you are only differentiating with respect to x.
Can this evaluate trigonometric functions?
Yes. This calculator's math parser supports standard mathematical operators, including sin(x), cos(x), tan(x), log(x), and exp(x). Just ensure you use * for multiplication (e.g., x * sin(y)).