Hyperbolic Tangent Calculator (tanh)

Compute tanh(x) and related hyperbolic function values with step-by-step breakdown.

Enter Value

Result

tanh(x)
--
Input x--
e^x--
e^(-x)--
sinh(x)--
cosh(x)--
tanh(x)--
coth(x)--
sech(x)--
csch(x)--

Step-by-Step Solution

tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x))

Understanding Hyperbolic Tangent

The hyperbolic tangent function, tanh(x), is one of the six hyperbolic functions. It is defined as the ratio of the hyperbolic sine to the hyperbolic cosine: tanh(x) = sinh(x)/cosh(x). Using exponential notation, tanh(x) = (ex - e-x) / (ex + e-x).

Key Properties

Definition

The ratio of hyperbolic sine to hyperbolic cosine.

tanh(x) = sinh(x) / cosh(x)

Exponential Form

Expressed using Euler's number e.

(e^x - e^-x) / (e^x + e^-x)

Range

The output is always between -1 and 1 (exclusive), approaching these as asymptotes.

Range: (-1, 1)

Odd Function

tanh(-x) = -tanh(x). The function is symmetric about the origin.

tanh(-x) = -tanh(x)

Derivative

The derivative of tanh(x) is sech squared of x.

d/dx tanh(x) = sech^2(x)

Identity

An important identity relating tanh and sech.

1 - tanh^2(x) = sech^2(x)

Applications of tanh

  • Activation function in neural networks and deep learning
  • Modeling catenary curves and hanging chains
  • Special relativity (rapidity)
  • Fluid dynamics and boundary layer theory
  • Signal processing and control theory