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 mathematics, physics, engineering, and more.

2. How Does the Calculator Work?

The calculator uses symbolic differentiation to compute derivatives:

Key Concepts:

• Power Rule: \( \frac{d}{dx}x^n = nx^{n-1} \)
• Product Rule: \( \frac{d}{dx}[uv] = u\frac{dv}{dx} + v\frac{du}{dx} \)
• Chain Rule: \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \)
• Common Derivatives:
  • \( \frac{d}{dx}\sin(x) = \cos(x) \)
  • \( \frac{d}{dx}\cos(x) = -\sin(x) \)
  • \( \frac{d}{dx}e^x = e^x \)
  • \( \frac{d}{dx}\ln(x) = \frac{1}{x} \)

3. Importance of Derivatives

Applications: Derivatives are used to find rates of change, slopes of curves, optimization problems, motion analysis, and in differential equations modeling real-world phenomena.

4. Using the Calculator

Tips: Enter mathematical functions using standard notation (x^2 for x², sin(x) for sine function). The calculator supports basic operations and common mathematical functions.

5. Frequently Asked Questions (FAQ)

Q1: What functions can I enter?
A: The calculator supports polynomials, trigonometric, exponential, and logarithmic functions with proper syntax.

Q2: Can I take derivatives with respect to variables other than x?
A: Yes, specify any variable in the "With Respect To" field (e.g., t for time derivatives).

Q3: Does this show step-by-step solutions?
A: This calculator provides the final result. For steps, consider a full computer algebra system.

Q4: How accurate are the results?
A: The calculator uses symbolic computation for exact derivatives, not numerical approximation.

Q5: Can I compute partial derivatives?
A: Yes, by specifying which variable to differentiate with respect to in a multivariable function.