6.9 Calculus of the Hyperbolic Functions

Hyperbolic Functions

Hyperbolic functions are defined using exponentials rather than the unit circle, which is what sets them apart from the regular trig functions you already know. They show up when modeling physical situations like the shape of a hanging cable (a catenary), signal processing, and certain differential equations. This section covers their definitions, derivatives, integrals, inverses, and a key real-world application.

Definitions of the Hyperbolic Functions

The six hyperbolic functions are built from combinations of $e^x$ and $e^{-x}$. The three primary ones are:

• $\sinh x = \frac{e^x - e^{-x}}{2}$ (hyperbolic sine)
• $\cosh x = \frac{e^x + e^{-x}}{2}$ (hyperbolic cosine)
• $\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ (hyperbolic tangent)

The remaining three are reciprocals, just like in regular trig:

• $\operatorname{csch} x = \frac{1}{\sinh x}$
• $\operatorname{sech} x = \frac{1}{\cosh x}$
• $\operatorname{coth} x = \frac{1}{\tanh x}$

Notice that $\cosh x$ is always positive and always $\geq 1$, while $\sinh x$ can be any real number. This is worth remembering because it affects domains later.

Derivatives of Hyperbolic Functions

The derivative rules here look a lot like regular trig derivatives, but watch the signs carefully. With regular trig, $\frac{d}{dx}\cos x = -\sin x$, but with hyperbolic functions there's no negative sign on the corresponding rule.

• $\frac{d}{dx} \sinh x = \cosh x$
• $\frac{d}{dx} \cosh x = \sinh x$ (no negative sign, unlike $\cos x$)
• $\frac{d}{dx} \tanh x = \operatorname{sech}^2 x$
• $\frac{d}{dx} \operatorname{csch} x = -\operatorname{csch} x \coth x$
• $\frac{d}{dx} \operatorname{sech} x = -\operatorname{sech} x \tanh x$
• $\frac{d}{dx} \operatorname{coth} x = -\operatorname{csch}^2 x$

You can verify any of these by substituting the exponential definitions and differentiating directly. For example, differentiating $\sinh x = \frac{e^x - e^{-x}}{2}$ gives $\frac{e^x + e^{-x}}{2} = \cosh x$.

Integrals of Hyperbolic Functions

Since integration reverses differentiation, many of these follow directly from the derivative table above.

• $\int \sinh x \, dx = \cosh x + C$
• $\int \cosh x \, dx = \sinh x + C$
• $\int \tanh x \, dx = \ln(\cosh x) + C$
• $\int \operatorname{coth} x \, dx = \ln|\sinh x| + C$
• $\int \operatorname{sech} x \, dx = \arctan(\sinh x) + C$
• $\int \operatorname{csch} x \, dx = \ln\left|\tanh \frac{x}{2}\right| + C$

The first two are straightforward. For $\int \tanh x \, dx$, rewrite it as $\int \frac{\sinh x}{\cosh x} \, dx$ and use the substitution $u = \cosh x$, so $du = \sinh x \, dx$. That turns it into $\int \frac{du}{u} = \ln|u| + C = \ln(\cosh x) + C$. You can drop the absolute value because $\cosh x > 0$ always. The same substitution idea works for $\operatorname{coth} x$.

Inverse Hyperbolic Functions

Each inverse hyperbolic function can be written as a logarithmic expression. This is because solving $y = \sinh x$ for $x$ (for example) turns into solving an equation in $e^x$, which yields a natural log.

• $\sinh^{-1} x = \ln\left(x + \sqrt{x^2 + 1}\right)$, domain: all real $x$
• $\cosh^{-1} x = \ln\left(x + \sqrt{x^2 - 1}\right)$, domain: $x \geq 1$
• $\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)$, domain: $-1 < x < 1$
• $\operatorname{csch}^{-1} x = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} + 1}\right)$, domain: $x \neq 0$
• $\operatorname{sech}^{-1} x = \ln\left(\frac{1}{x} + \sqrt{\frac{1}{x^2} - 1}\right)$, domain: $0 < x \leq 1$
• $\operatorname{coth}^{-1} x = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)$, domain: $|x| > 1$

Pay attention to the domain restrictions. They come from the ranges of the original hyperbolic functions. For instance, $\cosh x \geq 1$ for all $x$, so $\cosh^{-1}$ only accepts inputs $\geq 1$.

Derivatives of Inverse Hyperbolic Functions

These derivatives are important because they produce algebraic expressions involving square roots and rational functions. That means they show up as antiderivative results when you integrate certain algebraic forms.

• $\frac{d}{dx} \sinh^{-1} x = \frac{1}{\sqrt{x^2 + 1}}$
• $\frac{d}{dx} \cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}}$, for $x > 1$
• $\frac{d}{dx} \tanh^{-1} x = \frac{1}{1 - x^2}$, for $|x| < 1$
• $\frac{d}{dx} \operatorname{csch}^{-1} x = -\frac{1}{|x|\sqrt{x^2 + 1}}$
• $\frac{d}{dx} \operatorname{sech}^{-1} x = -\frac{1}{x\sqrt{1 - x^2}}$, for $0 < x < 1$
• $\frac{d}{dx} \operatorname{coth}^{-1} x = \frac{1}{1 - x^2}$, for $|x| > 1$

Notice that $\tanh^{-1} x$ and $\operatorname{coth}^{-1} x$ have the same derivative formula, $\frac{1}{1 - x^2}$, but they apply on different domains. When you see $\int \frac{1}{1 - x^2} \, dx$, the correct antiderivative depends on whether $|x| < 1$ (use $\tanh^{-1} x$) or $|x| > 1$ (use $\operatorname{coth}^{-1} x$).

Catenary Curves

A catenary is the curve formed when a uniform cable or chain hangs freely under its own weight between two fixed points. Its equation is:

$y = a \cosh\left(\frac{x}{a}\right)$

where $a$ is a constant determined by the ratio of the cable's tension to its weight per unit length. This shape minimizes the potential energy of the system, which is why it naturally appears as the equilibrium position.

Applications in engineering and architecture:

• Suspension bridges: The main cables approximate a catenary when supporting only their own weight. (When the cable supports a uniformly distributed deck load, the shape is actually a parabola, not a catenary. Exam problems sometimes test this distinction.)
• Power lines: Cables between utility poles form catenaries. Engineers use the catenary equation to calculate how much the cable sags, which determines tower height and spacing needed to maintain safe ground clearance.
• Arches and domes: An inverted catenary (flipped upside down) is an ideal arch shape because it carries loads in pure compression, minimizing internal bending stresses. The Gateway Arch in St. Louis is a well-known example of a weighted catenary shape.

In physics problems, the catenary is a classic application of calculus of variations and mechanical equilibrium. You may be asked to find the arc length of a catenary using the arc length integral, which simplifies nicely because $1 + \sinh^2 x = \cosh^2 x$ (the hyperbolic identity analogous to the Pythagorean identity).