1. What is Partial Derivative?

A partial derivative measures how a function changes as one variable changes, while keeping other variables constant. For a function f(x,y), the partial derivative with respect to x represents the rate of change of f in the x-direction.

2. How Does the Calculator Work?

The calculator uses the finite difference approximation:

Where:

• \( f(x,y) \) — Function value at point (x,y)
• \( f(x+h,y) \) — Function value at point (x+h,y)
• \( h \) — Small increment (typically 0.0001)

Explanation: This formula approximates the instantaneous rate of change by calculating the average rate of change over a very small interval h.

3. Importance of Partial Derivatives

Details: Partial derivatives are fundamental in multivariable calculus, optimization problems, physics (gradient, divergence), economics (marginal analysis), and machine learning (gradient descent).

4. Using the Calculator

Tips: Enter a mathematical function using x and y variables (e.g., "x^2 + y^2", "sin(x)*cos(y)"), specify the point (x,y) where you want to calculate the derivative, and choose a small h value for better accuracy.

5. Frequently Asked Questions (FAQ)

Q1: What is the ideal h value?
A: Typically 0.0001 to 0.000001. Too small may cause numerical instability, too large reduces accuracy.

Q2: Can I calculate ∂f/∂y?
A: This calculator computes ∂f/∂x. For ∂f/∂y, use the formula: [f(x,y+h) - f(x,y)]/h

Q3: What functions are supported?
A: Basic operations (+, -, *, /, ^), trigonometric functions (sin, cos, tan), and mathematical constants.

Q4: How accurate is this approximation?
A: The finite difference method provides an approximation. Accuracy improves with smaller h values, but extremely small h may introduce rounding errors.

Q5: What about second-order partial derivatives?
A: Second derivatives require more complex formulas and are not covered by this calculator.