Derivative of Tan⁻¹x

The derivative of tan⁻¹x is represented as d/dx (tan⁻¹x) or (tan⁻¹x)'. The value of this derivative is 1/(1+x²).

The inverse tangent function has a well-defined derivative, indicating it is differentiable across its domain.

Key concepts include:

Inverse Tangent Function: (tan⁻¹x is the inverse of the tangent function).

Chain Rule: Used for differentiating composite functions involving tan⁻¹x.

Domain Consideration: The function is differentiable for all real x.

The derivative of tan⁻¹x is expressed as: d/dx (tan⁻¹x) = 1/(1+x²) This formula is applicable for all real numbers x.

We can derive the derivative of tan⁻¹x using several proofs, employing trigonometric identities and differentiation rules. Here are some methods:

By First Principle

Using Chain Rule

We will demonstrate that the derivative of tan⁻¹x results in 1/(1+x²) using these methods:

By First Principle

The derivative of tan⁻¹x can be derived using the First Principle, expressing it as the limit of the difference quotient.

Let f(x) = tan⁻¹x. Its derivative is: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h Given f(x) = tan⁻¹x, write f(x + h) = tan⁻¹(x + h).

Substitute into the limit: f'(x) = limₕ→₀ [tan⁻¹(x + h) - tan⁻¹x] / h

Using the identity for tan⁻¹(a) - tan⁻¹(b), f'(x) = limₕ→₀ [1/(1+(x+h)x)(h)] = limₕ→₀ 1/(1+x²)

Hence, f'(x) = 1/(1+x²).

Using Chain Rule

To prove the differentiation of tan⁻¹x using the chain rule,

Consider y = tan⁻¹x.

Express tan⁻¹x = y then x = tan(y).

Differentiating both sides, dx/dy = sec²(y) Using the identity sec²(y) = 1 + tan²(y), dx/dy = 1 + x² Thus, dy/dx = 1/(1 + x²).

Therefore, the derivative of tan⁻¹x is 1/(1+x²).

Higher-order derivatives involve differentiating a function multiple times.

For example, the speed of a car changes (first derivative), and the rate at which that speed changes (second derivative).

Higher-order derivatives of tan⁻¹x provide deeper insights into its behavior.

First Derivative: f′(x) = 1/(1+x²).

Second Derivative: The derivative of f′(x) provides more information about the curvature.

Nth Derivative: fⁿ(x) represents the nth derivative of the function, indicating changes in the rate of change.

When x = 0, the derivative of tan⁻¹x = 1/(1+0²), which is 1. As x approaches ±∞, the derivative approaches 0, indicating the function levels off.

• Derivative: The derivative of a function indicates its rate of change with respect to a variable.

• Inverse Function: A function that reverses the effect of another function, such as tan⁻¹x.

• Chain Rule: A rule for differentiating composite functions.

• Limit: A fundamental concept in calculus used to define derivatives.

• Domain: The set of input values for which a function is defined.