## tanh x

### The hyperbolic tangent function, tanh x, is a mathematical function commonly used in calculus and trigonometry

The hyperbolic tangent function, tanh x, is a mathematical function commonly used in calculus and trigonometry. It is defined as:

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

Here, e represents the base of the natural logarithm, also known as Euler’s number. The function tanh x is defined for all real numbers.

The shape of the graph of the hyperbolic tangent function is similar to that of the standard tangent function (tan x), but it is symmetric with respect to the origin (0,0) and has a range from -1 to 1. As x approaches positive or negative infinity, the tanh x approaches ±1 respectively.

The hyperbolic tangent function has several important properties:

1. Odd function: tanh(-x) = -tanh(x). This means that the function is symmetric about the origin.

2. Hyperbolic identity: tanh(x) = (e^2x-1) / (e^2x+1). This implies that tanh(x) can also be expressed in terms of exponential functions.

3. Derivative: The derivative of tanh(x) with respect to x is given by sech^2(x), where sech(x) represents the hyperbolic secant function.

4. Relationship to other hyperbolic functions: tanh(x) is closely related to the hyperbolic sine and cosine functions by the identity: tanh(x) = sinh(x) / cosh(x).

The hyperbolic tangent function is utilized in various applications, including physics, engineering, and machine learning. It is commonly used to model nonlinear relationships and sigmoidal activation functions in neural networks.

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