Partial Differentiation Numerical Calculator
Calculate the partial derivative of a function f(x, y) at a specific point.
Use standard JS operators: * (multiply), / (divide), + (add), – (subtract), Math.pow(x, 2) or Math.sin(x)
x
y
Calculation Result
Understanding Partial Differentiation
Partial differentiation is a fundamental concept in multivariable calculus. Unlike ordinary differentiation, which deals with functions of a single variable, partial differentiation involves finding the derivative of a function with multiple variables with respect to one variable while keeping the others constant.
The Mathematical Definition
If we have a function f(x, y), the partial derivative with respect to x is denoted as ∂f/∂x or fₓ. It is defined by the limit:
fₓ(x, y) = lim(h -> 0) [f(x + h, y) - f(x, y)] / h
This tells us the rate of change of the function along the x-axis at a specific point, treating the y-coordinate as a fixed value.
How to Use This Calculator
This tool uses a numerical approximation method (the difference quotient with a very small step) to determine the gradient at your chosen point. To use it:
- Function Input: Enter your expression using JavaScript syntax. For example,
x^2should bex*xand3xyshould be3*x*y. - Variable selection: Choose whether you want the slope relative to the horizontal (x) or depth (y) axis.
- Point: Specify the exact coordinates (x, y) where you want the slope calculated.
Real-World Example
Physics – Ideal Gas Law: Consider the pressure
P(V, T) = nRT / V. To find how pressure changes with temperature while volume is constant, we calculate the partial derivative ∂P/∂T. This is essential in thermodynamics for understanding engine cycles and atmospheric changes.
Common Notations
- ∂ (Del): The specific symbol used for partial derivatives to distinguish them from the "d" used in ordinary calculus.
- Subscript: Writing
fₓorfᵧis a shorthand way to indicate the variable of differentiation. - Gradient Vector: The vector containing all partial derivatives
∇f = (fₓ, fᵧ), which points in the direction of steepest ascent.
f with respect to " + variable + " at point (" + x + ", " + y + ") is approximately:" + finalResult + "";
} catch (error) {
alert("Error in function expression. Please ensure you are using valid math syntax (e.g., use * for multiplication).");
console.error(error);
}
}