Calculus/Derivatives of Hyperbolic Functions

The hyperbolic functions are functions that are related to the trigonometric functions, largely due to the consequences of their definitions. While not necessarily related to the triangular or circular definitions of the trigonometric functions, their identities make them look quite similar.

Nevertheless, these functions are important in calculus because they form a large subset of solutions to some problems in calculus and are important in many areas of applied mathematics.

Introducing the Hyperbolic Functions

A ray through the unit hyperbola Template:Math in the point Template:Math, where $a$ is twice the area between the ray, the hyperbola, and the $x$-axis. For points on the hyperbola below the $x$-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).

Section Template:Calculus/map page defines hyperbolic functions according to the parametric definition, similar to trigonometric functions.

That is, rotating a ray from the direction of the positive half of the $x$-axis by an angle $\theta$ (counterclockwise for $\theta >0,$ and clockwise for $\theta <0$) yields intersection points of this ray with the unit hyperbola: $A=(x_A,y_A)$.

It is defined that: Template:Equation Template:Equation Template:Equation

where $a$ is twice the area between the ray, the hyperbola, and the $x$-axis. As a consequence of these definitions, it becomes clear

Keep in mind that similar to $\sec (x)$, $\csc (x)$, and $\cot (x)$, the definitions of the hyperbolic versions are as follows: Template:Equation Template:Equation Template:Equation

These definitions are limited in that applying calculus to this would be difficult without additional tools. Using the exponential definition of hyperbolic functions (not defined in Section Template:Calculus/map page) allow us to more easily find derivatives, the goal of this section.

Properties of the Hyperbolic Functions

Proof: By equation Template:Eqnref, $\sinh x=\frac{e^x-e^{-x}}{2}$. If a function is odd, then $f(-x)=-f(x)$.

$\begin{array}{cc}\sinh (-x)& =\frac{e^{(-x)}-e^{-(-x)}}{2}\\ & =\frac{e^{-x}-e^x}{2}\\ & =-\frac{e^x-e^{-x}}{2}=-\sinh (x)◼\end{array}$

Proof: By equation Template:Eqnref, $\cosh x=\frac{e^x+e^{-x}}{2}$. If a function is even, then $f(-x)=f(x)$.

$\begin{array}{cc}\cosh (-x)& =\frac{e^{(-x)}+e^{-(-x)}}{2}\\ & =\frac{e^{-x}+e^x}{2}=\frac{e^x+e^{-x}}{2}=\cosh (x)◼\end{array}$

Proof: Using equations Template:Eqnref and Template:Eqnref, $\cosh x=\frac{e^x+e^{-x}}{2}$,

$\begin{array}{cc}\sinh (2x)& =\frac{e^{(2x)}+e^{-(2x)}}{2}\\ & =\frac{e^{4x}-1}{2e^{2x}}\\ & =\frac{(e^{2x}-1)(e^{2x}+1)}{2e^{2x}}\end{array}$

Notice that $\sinh (x)\cosh (x)=\frac{e^{4x}-1}{4e^{2x}}$. Multiplying this by two gives us $\sinh (2x)$. Hence,

$\sinh (2x)=2\sinh (x)\cosh (x)◼$

Proof:

Derivatives of the Hyperbolic Functions

Proof:

Proof: