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 across physics, engineering, economics, and more.

2. How Does the Calculator Work?

The calculator uses the limit definition of the derivative:

Where:

• \( f(x) \) — The input function
• \( h \) — An infinitesimally small change in x
• \( f(x+h) - f(x) \) — The change in function value

Explanation: The derivative measures the instantaneous rate of change or the slope of the tangent line at any point on the function's graph.

3. Importance of Derivatives

Details: Derivatives are essential for understanding rates of change in physical systems, optimizing functions, solving differential equations, and modeling real-world phenomena.

4. Using the Calculator

Tips: Enter a mathematical function using standard notation (e.g., x^2 for x squared, sin(x) for sine of x). Specify the variable of differentiation (default is x).

5. Frequently Asked Questions (FAQ)

Q1: What functions can the calculator handle?
A: The calculator can handle polynomials, trigonometric functions (sin, cos), exponential functions, and basic combinations of these.

Q2: How accurate is the calculator?
A: The calculator provides symbolic differentiation, which is mathematically exact for supported functions.

Q3: What's the difference between derivative and differential?
A: The derivative is the rate of change (a function), while the differential is the actual change in the function's value for a small change in input.

Q4: Can I calculate higher-order derivatives?
A: This calculator computes first derivatives. For second or higher derivatives, you would need to apply the derivative operation multiple times.

Q5: What are some common derivative rules?
A: Key rules include the power rule (d/dx x^n = nx^(n-1)), product rule, quotient rule, and chain rule for composite functions.