Derivative Calculator

Derivative Rules:

\[ \frac{d}{dx}(x^n) = nx^{n-1} \] \[ \frac{d}{dx}(e^x) = e^x \] \[ \frac{d}{dx}(\ln x) = \frac{1}{x} \] \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin x \]

Derivative Result:

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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 Does the Calculator Work?

The calculator uses symbolic differentiation rules to compute derivatives:

\[ \frac{d}{dx}(x^n) = nx^{n-1} \] \[ \frac{d}{dx}(e^x) = e^x \] \[ \frac{d}{dx}(\ln x) = \frac{1}{x} \] \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin x \]

Note: This is a simplified demonstration. A full implementation would require a proper symbolic math library.

3. Importance of Derivatives

Applications: Derivatives are used to find rates of change, optimize functions, solve differential equations, and model real-world phenomena like motion, growth, and decay.

4. Using the Calculator

Tips: Enter mathematical functions using standard notation (e.g., x^2 for x squared, sin(x) for sine of x). Select the variable you want to differentiate with respect to.

5. Frequently Asked Questions (FAQ)

Q1: What functions can this calculator handle?
A: This demo version handles basic polynomials, exponential, logarithmic, and trigonometric functions.

Q2: Can I compute partial derivatives?
A: Yes, by selecting a variable other than x (like y or t) for "With Respect To".

Q3: What about more complex functions?
A: A full implementation would handle product rule, quotient rule, and chain rule for composite functions.

Q4: Can I see step-by-step solutions?
A: This demo doesn't show steps, but advanced versions could display the differentiation process.

Q5: Is this calculator suitable for educational use?
A: While helpful for verification, students should learn to compute derivatives manually first.