What is the derivative of tanh ( x )?

The derivative of the hyperbolic tangent function, denoted as tanh(x), can be calculated using the definition of derivatives and properties of hyperbolic functions.

Derivative of tanh(x)

The formula for the derivative of tanh(x) is:

Formula

tanh'(x) = sech²(x)

Understanding sech(x)

Here, sech(x) is the hyperbolic secant function, defined as:

sech(x) = 1 / cosh(x)

Summary

• The derivative of tanh(x) is sech²(x).
• This means that as x changes, the rate of change of tanh(x) is determined by the square of the hyperbolic secant of x.

In practical terms, this derivative helps in understanding the behavior of the tanh function in calculus and various applications in mathematics and engineering.