Partial Derivative Calculator

Partial Derivative Calculator for Quadratic Functions

This calculator helps you evaluate the partial derivatives of a two-variable quadratic function at a specific point. The general form of the function is:

f(x, y) = A*x2 + B*y2 + C*x*y + D*x + E*y + F

The partial derivatives are:

• ∂f/∂x = 2*A*x + C*y + D
• ∂f/∂y = 2*B*y + C*x + E

Function Coefficients:

Evaluation Point:

function calculatePartialDerivatives() { var A = parseFloat(document.getElementById('coeffA').value); var B = parseFloat(document.getElementById('coeffB').value); var C = parseFloat(document.getElementById('coeffC').value); var D = parseFloat(document.getElementById('coeffD').value); var E = parseFloat(document.getElementById('coeffE').value); var F = parseFloat(document.getElementById('coeffF').value); var x = parseFloat(document.getElementById('evalX').value); var y = parseFloat(document.getElementById('evalY').value); if (isNaN(A) || isNaN(B) || isNaN(C) || isNaN(D) || isNaN(E) || isNaN(F) || isNaN(x) || isNaN(y)) { document.getElementById('dfDxResult').innerHTML = 'Please enter valid numbers for all fields.'; document.getElementById('dfDyResult').innerHTML = "; document.getElementById('fXYResult').innerHTML = "; return; } // Calculate partial derivatives var dfDx = (2 * A * x) + (C * y) + D; var dfDy = (2 * B * y) + (C * x) + E; // Calculate the function value itself (optional but good for context) var fXY = (A * Math.pow(x, 2)) + (B * Math.pow(y, 2)) + (C * x * y) + (D * x) + (E * y) + F; document.getElementById('dfDxResult').innerHTML = dfDx.toFixed(4); document.getElementById('dfDyResult').innerHTML = dfDy.toFixed(4); document.getElementById('fXYResult').innerHTML = fXY.toFixed(4); }

Understanding Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. When you have a function with two or more independent variables, a partial derivative measures how the function changes with respect to one of those variables, while holding all other variables constant.

Imagine a mountain represented by a function f(x, y), where x and y are your coordinates on a map, and f(x, y) is the altitude. The partial derivative ∂f/∂x tells you the slope of the mountain if you walk directly east or west (changing only x, keeping y constant). Similarly, ∂f/∂y tells you the slope if you walk directly north or south (changing only y, keeping x constant).

The Quadratic Function Example

This calculator focuses on a specific type of function: a quadratic function of two variables, given by f(x, y) = A*x2 + B*y2 + C*x*y + D*x + E*y + F. This form is common in various fields, including:

• Optimization: Finding maximum or minimum points of a surface.
• Physics: Describing potential energy fields or stress distributions.
• Economics: Modeling production functions or utility functions.

For this specific function, the rules of differentiation are applied to each term, treating the other variable as a constant:

• To find ∂f/∂x, we differentiate f(x, y) with respect to x, treating y as a constant.
  • d/dx (A*x2) = 2*A*x
  • d/dx (B*y2) = 0 (since y is constant)
  • d/dx (C*x*y) = C*y (since C and y are constants)
  • d/dx (D*x) = D
  • d/dx (E*y) = 0
  • d/dx (F) = 0
  Summing these gives: ∂f/∂x = 2*A*x + C*y + D
• To find ∂f/∂y, we differentiate f(x, y) with respect to y, treating x as a constant.
  • d/dy (A*x2) = 0
  • d/dy (B*y2) = 2*B*y
  • d/dy (C*x*y) = C*x
  • d/dy (D*x) = 0
  • d/dy (E*y) = E
  • d/dy (F) = 0
  Summing these gives: ∂f/∂y = 2*B*y + C*x + E

How to Use the Calculator

1. Identify your function: Ensure your function is in the form f(x, y) = A*x2 + B*y2 + C*x*y + D*x + E*y + F.
2. Input Coefficients: Enter the numerical values for A, B, C, D, E, and F into the respective fields. If a term is missing (e.g., no x2 term), its coefficient is 0.
3. Specify Evaluation Point: Enter the specific x and y values at which you want to find the partial derivatives.
4. Calculate: Click the "Calculate Partial Derivatives" button.
5. View Results: The calculator will display the value of ∂f/∂x and ∂f/∂y at your specified point, along with the function's value f(x, y) at that point.

Example Calculation

Let's consider the function f(x, y) = 2x2 + 3y2 - 4xy + 5x - 2y + 10. We want to find the partial derivatives at the point (x=1, y=2).

From the function, we identify the coefficients:

• A = 2
• B = 3
• C = -4
• D = 5
• E = -2
• F = 10

The evaluation point is x = 1 and y = 2.

Using the formulas:

• ∂f/∂x = 2*A*x + C*y + D
• ∂f/∂x = 2*(2)*(1) + (-4)*(2) + 5
• ∂f/∂x = 4 - 8 + 5 = 1

• ∂f/∂y = 2*B*y + C*x + E
• ∂f/∂y = 2*(3)*(2) + (-4)*(1) + (-2)
• ∂f/∂y = 12 - 4 - 2 = 6

And the function value:

• f(x, y) = A*x2 + B*y2 + C*x*y + D*x + E*y + F
• f(1, 2) = 2*(1)2 + 3*(2)2 - 4*(1)*(2) + 5*(1) - 2*(2) + 10
• f(1, 2) = 2*(1) + 3*(4) - 8 + 5 - 4 + 10
• f(1, 2) = 2 + 12 - 8 + 5 - 4 + 10 = 17

So, at the point (1, 2), the rate of change with respect to x is 1, and with respect to y is 6. The function's value at that point is 17.

You can input these values into the calculator above to verify the results.