Derivative Rule Of Inverse Hyperbolic Tangent Function - Math Doubts

Trang chủ » D/dx Tanh^-1 X » Derivative Rule Of Inverse Hyperbolic Tangent Function - Math Doubts

Có thể bạn quan tâm

  • Algebra
  • Trigonometry
  • Geometry
  • Calculus
Math Doubts Derivative Rule of Inverse Hyperbolic Tangent function
  • Math Doubts
  • Differential calculus
  • Differentiation
  • Rules
  • Inverse Hyperbolic functions

Formula

$\dfrac{d}{dx}{\,\tanh^{-1}{x}}$ $\,=\,$ $\dfrac{1}{1-x^2}$

Introduction

The inverse hyperbolic tangent is written in function form as $\tanh^{-1}{(x)}$ or $\operatorname{arctanh}{(x)}$ if the literal $x$ represents a variable. The differentiation of the inverse hyperbolic tan function with respect to $x$ is written in the following mathematical forms.

$(1).\,\,\,$ $\dfrac{d}{dx}{\, (\tanh^{-1}{x})}$

$(2).\,\,\,$ $\dfrac{d}{dx}{\, (\operatorname{arctanh}{x})}$

In simple form, the derivative of inverse hyperbolic tan function is written as $(\tanh^{-1}{x})’$ or $(\operatorname{arctanh}{x})’$ mathematically in differential calculus.

The differentiation of hyperbolic inverse tangent function with respect to $x$ is equal to multiplicative inverse of difference of $x$ squared from one.

$\implies$ $\dfrac{d}{dx}{\, \tanh^{-1}{x}}$ $\,=\,$ $\dfrac{1}{1-x^2}$

Other forms

The derivative of inverse hyperbolic tangent function can also be expressed in any variable in mathematics.

Example

$(1) \,\,\,$ $\dfrac{d}{dl}{\, \tanh^{-1}{l}}$ $\,=\,$ $\dfrac{1}{1-l^2}$

$(2) \,\,\,$ $\dfrac{d}{dq}{\, \tanh^{-1}{q}}$ $\,=\,$ $\dfrac{1}{1-q^2}$

$(3) \,\,\,$ $\dfrac{d}{dy}{\, \tanh^{-1}{y}}$ $\,=\,$ $\dfrac{1}{1-y^2}$

Proof

Learn how to prove differentiation rule of inverse hyperbolic tangent function from the first principle of differentiation.

Learn more
Latest Math Concepts
Jan 12, 2026
(a−b)² Formula Explained with Expansion, Proof, Examples & Uses
Jan 09, 2026
What are Factors of 2? Positive, Negative, & Factor Pairs
Jan 08, 2026
What is a Negative Factor of a number?
Oct 13, 2025
What is a Trigonometric ratio?
Sep 18, 2025
What is a Point in geometry?
Sep 13, 2025
What is a Factor in Higher mathematics?
Latest Math Questions
Jan 18, 2026
Evaluate ∫sec(2x) dx
Jan 02, 2026
Prove $\log{(1+2+3)}$ $\,=\,$ $\log{(1)}$ $+$ $\log{(2)}$ $+$ $\log{(3)}$
Oct 26, 2025
Evaluate $5\cos{x}$ $-$ $12\sin{x}$, if $5\sin{x}$ $+$ $12\cos{x}$ $=$ $13$
Oct 11, 2025
Evaluate $\displaystyle \large \lim_{x \,\to\, 0}{\normalsize \dfrac{\sqrt{1-\cos{2x}}}{x}}$
Oct 03, 2025
Evaluate $\dfrac{1}{\log_{2}{30}}$ $+$ $\dfrac{1}{\log_{3}{30}}$ $+$ $\dfrac{1}{\log_{5}{30}}$
Jul 19, 2025
Evaluate $\displaystyle \large \lim_{x \,\to\, 2}{\normalsize (x^3-5x+6)}$ by Direct substitution

Từ khóa » D/dx Tanh^-1 X