First Second And Third Derivative Calculator

Derivative Notation:

\[ f'(x) = \frac{d}{dx}f(x), \quad f''(x) = \frac{d^2}{dx^2}f(x), \quad f'''(x) = \frac{d^3}{dx^3}f(x) \]

First Derivative f'(x):

Second Derivative f''(x):

Third Derivative f'''(x):

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1. What Are Derivatives?

Derivatives represent the rate of change of a function with respect to its variable. The first derivative measures instantaneous rate of change, the second derivative measures the rate of change of the first derivative (concavity), and the third derivative measures the rate of change of the second derivative.

2. How The Calculator Works

The calculator computes derivatives using symbolic differentiation rules:

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

The calculator applies these rules recursively to find higher-order derivatives.

3. Importance of Derivatives

Applications: Derivatives are fundamental in physics (velocity, acceleration), economics (marginal analysis), engineering (optimization), and many other fields where rate of change analysis is required.

4. Using The Calculator

Instructions: Enter a mathematical function using standard notation (x^2 for x², sin(x), cos(x), exp(x), etc.). The calculator will compute the first, second, and third derivatives.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can the calculator handle?
A: The calculator can handle polynomial, trigonometric, exponential, and logarithmic functions.

Q2: How accurate are the computed derivatives?
A: The calculator provides exact symbolic derivatives, not numerical approximations.

Q3: Can I use variables other than x?
A: Currently, the calculator only supports the variable 'x' for differentiation.

Q4: What if my function has multiple terms?
A: The calculator follows the sum rule: derivative of f(x) + g(x) is f'(x) + g'(x).

Q5: Does the calculator show the step-by-step process?
A: This version provides the final results. Future versions may include step-by-step solutions.