1. What is Instantaneous Rate of Change?

The Instantaneous Rate of Change (IROC) is the rate at which a quantity changes at a specific point, represented mathematically as the derivative of a function at a point. It's the limit of the average rate of change as the interval approaches zero.

2. How Does the Calculator Work?

The calculator uses the numerical approximation of the derivative:

Where:

• \( f(x) \) — The input function
• \( x \) — The point at which to calculate the rate of change
• \( h \) — A very small number (approaching zero)

Explanation: This calculator provides a numerical approximation of the derivative by using a very small h value (default: 0.0001).

3. Importance of IROC Calculation

Details: IROC is fundamental in calculus and physics, representing concepts like velocity (rate of change of position), acceleration (rate of change of velocity), and many other rates in science and engineering.

4. Using the Calculator

Tips:

• Enter functions using standard mathematical notation (e.g., "x^2 + 3*x - 5")
• Use a very small h value (0.0001 or smaller) for better approximations
• For trigonometric functions, use sin(x), cos(x), tan(x) etc.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between average and instantaneous rate of change?
A: Average rate of change is over an interval, while instantaneous is at a single point (the derivative).

Q2: How small should h be?
A: Smaller h gives better approximation but too small may cause floating-point errors. 0.0001 is typically good.

Q3: What functions can I enter?
A: Basic arithmetic, exponents (x^2 or pow(x,2)), and PHP-compatible math functions (sin, cos, exp, log, etc.).

Q4: Why is my result slightly different from exact derivative?
A: This is a numerical approximation. The exact derivative would require symbolic differentiation.

Q5: Can I use this for multivariable functions?
A: No, this calculates partial derivative with respect to x only. For multivariable functions, you'd need partial derivatives.