Higher Derivatives of the Tangent and Inverse Tangent Functions and Chebyshev Polynomials

TL;DR

This paper derives formulas for higher derivatives of tangent, hyperbolic tangent, inverse tangent, and inverse hyperbolic tangent functions, linking them to Chebyshev polynomials and providing new polynomial representations.

Contribution

It introduces new formulas for higher derivatives of inverse tangent functions expressed via Chebyshev polynomials, connecting special functions with polynomial theory.

Findings

Derived formulas for higher derivatives of tangent and hyperbolic tangent functions.

Established polynomial representations for derivatives of inverse tangent functions.

Connected derivatives of inverse tangent functions to Chebyshev polynomials of the second kind.

Abstract

The higher derivatives of the tangent and hyperbolic tangent functions are determined. Formulas for the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions as polynomials are stated and proved. Using another formula for the higher derivatives of the inverse tangent function from literature, two known formulas for the Chebyshev polynomials of the first and second kind are proved. From these formulas the higher derivatives of the inverse tangent and inverse hyperbolic tangent functions in terms of the Chebyshev polynomial of the second kind are provided.