1. What is a Derivative at a Point?

The derivative of a function at a point represents the instantaneous rate of change of the function at that specific point. Geometrically, it's the slope of the tangent line to the function's graph at that point.

2. How Does the Calculator Work?

The calculator uses the limit definition of derivative:

Where:

• \( f \) — The input function
• \( a \) — The point at which to calculate the derivative
• \( h \) — A small number approaching zero

Explanation: The calculator approximates the limit by using a very small value of h (default 0.0001) to compute the difference quotient.

3. Importance of Derivatives

Details: Derivatives are fundamental in calculus with applications in physics, engineering, economics, and more. They describe rates of change in motion, growth, decay, and optimization problems.

4. Using the Calculator

Tips:

• Enter functions using standard mathematical notation (e.g., "x^2" for x squared)
• Use common functions: sin(x), cos(x), tan(x), exp(x), ln(x), log(x)
• For better accuracy, use smaller h values (but not too small to avoid floating-point errors)

5. Frequently Asked Questions (FAQ)

Q1: Why is my result not perfectly accurate?
A: This uses numerical approximation. For exact results, use symbolic differentiation.

Q2: What functions can I input?
A: Polynomials, trigonometric, exponential, and logarithmic functions are supported.

Q3: How small should h be?
A: Typically 0.0001 is good. Too small may cause floating-point precision errors.

Q4: Can I calculate higher-order derivatives?
A: This calculator only finds first derivatives. For second derivatives, apply the calculator twice.

Q5: Why do I get errors with some functions?
A: Ensure proper syntax and that the function is defined at point a.