Physlib.Mathematics.Trigonometry.Tanh

The derivative of the real hyperbolic tangent function $\tanh$ is given by $\frac{d}{dx} \tanh x = 1 - \tanh^2 x$ for all $x \in \mathbb{R}$.

For any natural number $n$, the hyperbolic tangent function $\tanh: \mathbb{R} \to \mathbb{R}$ is $n$-times continuously differentiable.

For any natural number $n$, there exists a polynomial $P$ with real coefficients such that for all $x \in \mathbb{R}$, the $n$-th derivative of the hyperbolic tangent function is given by $\frac{d^n}{dx^n} \tanh x = P(\tanh x)$.

For any polynomial $P$ with real coefficients and any real numbers $a$ and $b$, there exists a real constant $M$ such that for all $x$ in the closed interval $[a, b]$, the absolute value of the polynomial evaluated at $x$ satisfies $|P(x)| \le M$.

For any polynomial $P$ with real coefficients, there exists a real constant $C$ such that for all real numbers $x$, the absolute value of the polynomial evaluated at $\tanh x$ satisfies $|P(\tanh x)| \le C$.

For any natural number $n$, there exists a real constant $C$ such that for all $x \in \mathbb{R}$, the absolute value of the $n$-th derivative of the hyperbolic tangent function satisfies $\left| \frac{d^n}{dx^n} \tanh x \right| \le C$.

The hyperbolic tangent function $\tanh: \mathbb{R} \to \mathbb{R}$ is infinitely continuously differentiable, meaning it is of class $C^\infty$.

The hyperbolic tangent function $\tanh: \mathbb{R} \to \mathbb{R}$ has temperate growth. In the context of Schwartz space theory, this implies that $\tanh$ is a smooth function whose derivatives are all bounded by polynomials.

For any natural number $n$, the $n$-th derivative of the hyperbolic tangent function $\tanh: \mathbb{R} \to \mathbb{R}$ is differentiable on $\mathbb{R}$.

For any natural number $n$, any real number $x$, and a real constant $\kappa$, the norm of the $n$-th Fréchet derivative of the function $f(x) = \tanh(\kappa x)$ is equal to the absolute value of its $n$-th scalar derivative at $x$. That is, $$\| \text{D}^n (\tanh(\kappa \cdot))(x) \| = | (\tanh(\kappa \cdot))^{(n)}(x) |$$ where $\text{D}^n$ denotes the $n$-th Fréchet derivative and $^{(n)}$ denotes the $n$-th scalar derivative.

For any natural number $n$, real constant $\kappa$, and real number $x$, the $n$-th derivative of the scaled hyperbolic tangent function $f(x) = \tanh(\kappa x)$ is given by $$\frac{d^n}{dx^n} \tanh(\kappa x) = \kappa^n \tanh^{(n)}(\kappa x)$$ where $\tanh^{(n)}$ denotes the $n$-th derivative of the hyperbolic tangent function.

For any real constant $\kappa$, the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \tanh(\kappa x)$ has temperate growth. A function is said to have temperate growth (or is slowly increasing) if it is $C^\infty$ and each of its derivatives is bounded by a polynomial.