Hyperbolic Functions sinh, cosh, tanh: Graphs, Derivatives, Integrals

The hyperbolic functions \sinh x, \cosh x and \tanh x are combinations of e^x and e^{-x} that behave remarkably like the trig functions. They share almost the same identities and derivative rules, but with a few sign changes, and they describe real shapes such as a hanging cable.

Definitions

Adding the first two gives \cosh x + \sinh x = e^x, so they split the natural exponential function into an even part (\cosh) and an odd part (\sinh). The reciprocals are defined just like their trig cousins: \operatorname{sech} x = \frac{1}{\cosh x}, \operatorname{csch} x = \frac{1}{\sinh x} and \coth x = \frac{1}{\tanh x}.

Graphs of sinh, cosh and tanh

\tanh x has horizontal asymptotes y = 1 and y = -1, since \tanh x \to 1 as x \to \infty and \tanh x \to -1 as x \to -\infty.

Hyperbolic identities

The central identity comes straight from the definitions:

Compare it with \cos^2 x + \sin^2 x = 1. Other useful identities:

• 1 - \tanh^2 x = \operatorname{sech}^2 x
• \sinh(2x) = 2\sinh x\cosh x
• \cosh(2x) = \cosh^2 x + \sinh^2 x

Derivatives of hyperbolic functions

Each follows from \frac{d}{dx}e^{-x} = -e^{-x}. For example,

For \tanh x, the quotient rule gives \frac{\cosh^2 x - \sinh^2 x}{\cosh^2 x} = \frac{1}{\cosh^2 x} = \operatorname{sech}^2 x.

Integrals of hyperbolic functions

The \tanh integral uses u = \cosh x: \int\frac{\sinh x}{\cosh x}\,dx = \ln|\cosh x| + C, and the absolute value can be dropped because \cosh x \ge 1.

Worked examples

Example 1: chain rule. \frac{d}{dx}\cosh(x^2) = 2x\sinh(x^2).

Example 2: evaluating. Find \cosh(\ln 2). Since e^{\ln 2} = 2 and e^{-\ln 2} = \frac{1}{2},

Example 3: a definite integral. \int_0^{1}\cosh x\,dx = \sinh 1 - \sinh 0 = \frac{e - e^{-1}}{2} \approx 1.175.

Example 4: u-substitution. \int\sinh(3x)\,dx = \frac{1}{3}\cosh(3x) + C.

Common mistakes

1. Copying the trig minus sign. \frac{d}{dx}\cosh x = \sinh x, not -\sinh x.
2. Using the trig identity. It is \cosh^2 x - \sinh^2 x = 1, with a minus sign.
3. Expecting periodicity. Hyperbolic functions never repeat. \sinh and \cosh grow like \frac{e^x}{2} for large x.
4. Confusing \sinh x with \sin x. They are different functions; \sinh 0 = 0 like \sin 0, but \sinh x is unbounded.

Where hyperbolic functions appear

A hanging cable takes the shape of a catenary, y = a\cosh\frac{x}{a}, and its arc length is easy to compute because 1 + \sinh^2 x = \cosh^2 x. The substitution x = \sinh t is an alternative to trig substitution for integrals with \sqrt{1 + x^2}. Hyperbolic functions also have tidy Taylor series: \sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots, the sine series with every sign positive. For background on the exponential derivatives behind all of this, see derivatives of exponential and logarithmic functions.