1. What is the Derivative of Arctan?

The derivative of arctan(x) (inverse tangent function) is given by the formula 1/(1 + x²). This represents the rate of change of the arctangent function with respect to x.

2. How Does the Calculator Work?

The calculator uses the derivative formula:

Where:

• \( x \) — Input value (unitless)

Explanation: The calculator computes the derivative of arctan(x) at any given point x using the standard mathematical formula.

3. Importance of Derivative Calculation

Details: Calculating derivatives is fundamental in calculus and has applications in physics, engineering, economics, and various scientific fields where rate of change analysis is required.

4. Using the Calculator

Tips: Enter any real number value for x. The calculator will compute the derivative of arctan(x) at that point.

5. Frequently Asked Questions (FAQ)

Q1: What is the domain of arctan(x)?
A: The arctan function is defined for all real numbers, so x can be any real value.

Q2: What is the range of the derivative?
A: The derivative 1/(1 + x²) ranges between 0 and 1, with maximum value 1 at x = 0.

Q3: How is this derivative derived?
A: Using implicit differentiation and trigonometric identities, starting from tan(y) = x and differentiating both sides.

Q4: What are practical applications?
A: Used in optimization problems, physics (angular calculations), signal processing, and computer graphics.

Q5: Can this be used for complex numbers?
A: This calculator handles real numbers only. Complex number derivatives require different treatment.