1. What is a Derivative?

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input variable. It's a fundamental concept in calculus with applications in physics, engineering, economics, and more.

2. How Differentiation Works

The derivative is defined as the limit:

Where:

• \( f(x) \) — The original function
• \( f'(x) \) — The derivative of f(x)
• \( h \) — An infinitesimally small change in x

Explanation: The derivative gives the slope of the tangent line to the function at any point, representing the instantaneous rate of change.

3. Common Derivative Rules

Basic Rules:

• Power Rule: \( \frac{d}{dx}x^n = nx^{n-1} \)
• Constant Rule: \( \frac{d}{dx}c = 0 \)
• Sum Rule: \( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \)
• Product Rule: \( \frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \)
• Chain Rule: \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \)

4. Using the Calculator

Tips: Enter mathematical expressions using standard notation (e.g., x^2 for x squared, sin(x) for sine function). The calculator will compute the derivative with respect to your chosen variable.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can this calculator differentiate?
A: The calculator can handle polynomials, trigonometric, exponential, and logarithmic functions.

Q2: Does the calculator show step-by-step solutions?
A: This version shows the final result. Advanced versions may include step-by-step differentiation.

Q3: Can I compute higher-order derivatives?
A: Currently this calculates first derivatives. Future versions may include second and third derivatives.

Q4: How accurate are the results?
A: The calculator uses symbolic differentiation for precise mathematical results.

Q5: Can I use this for implicit differentiation?
A: This calculator currently handles explicit functions. Implicit differentiation requires special notation.