Derivatives of Hyperbolic Functions, calculus lesson

Hold a chain or a cable loosely by its two ends and let it hang. The curve it settles into looks like a parabola but is not one. It is a scaled copy of the graph of the hyperbolic cosine. The hyperbolic functions are built from $e^x$ and $e^{-x}$ , and they behave so much like sine and cosine that their formulas are easy to remember once you know where the signs differ.

Key terms

Exponential definitions

Definition For every real $x$ , the hyperbolic sine and hyperbolic cosine are

\sinh x = \frac{e^x - e^{-x}}{2}, \qquad \cosh x = \frac{e^x + e^{-x}}{2}.

In plain words The function $\cosh x$ is the even part of $e^x$ and $\sinh x$ is its odd part. Adding them gives back $e^x$ .

Example At $x = \ln 2$ we have $e^x = 2$ and $e^{-x} = \frac{1}{2}$ , so

\sinh(\ln 2) = \frac{2 - 1/2}{2} = \frac{3}{4}, \qquad \cosh(\ln 2) = \frac{2 + 1/2}{2} = \frac{5}{4}.

Hyperbolic sine

Definition The function $\sinh$ is odd, defined and differentiable on $\mathbb{R}$ , with range $\mathbb{R}$ , and

\frac{d}{dx} \sinh x = \frac{e^x + e^{-x}}{2} = \cosh x.

In plain words The slope of $\sinh$ is $\cosh$ , which is always at least 1, so $\sinh$ is increasing everywhere.

Example At $x = 0$ the value is $\sinh 0 = 0$ and the slope is $\cosh 0 = 1$ , so the tangent line there is $y = x$ .

Hyperbolic cosine

Definition The function $\cosh$ is even, defined and differentiable on $\mathbb{R}$ , with range $[1, \infty)$ , and

\frac{d}{dx} \cosh x = \frac{e^x - e^{-x}}{2} = \sinh x.

In plain words Unlike cosine, there is no minus sign here. Since $\sinh x$ is negative for $x < 0$ and positive for $x > 0$ , the graph of $\cosh$ falls, bottoms out at height 1 when $x = 0$ , then rises.

Example By the chain rule

\frac{d}{dx} \cosh(3x) = 3 \sinh(3x).

At $x = \ln 2$ the slope of $\cosh x$ itself is $\sinh(\ln 2) = \frac{3}{4}$ .

Identities

Definition For every real $x$ ,

\cosh^2 x - \sinh^2 x = 1, \qquad \sinh(2x) = 2 \sinh x \cosh x, \qquad \cosh(2x) = \cosh^2 x + \sinh^2 x,

and, dividing the first by $\cosh^2 x$ ,

1 - \tanh^2 x = \operatorname{sech}^2 x.

Each follows by expanding the exponential definitions.

In plain words The point $(\cosh t, \sinh t)$ lies on the hyperbola $x^2 - y^2 = 1$ , just as $(\cos t, \sin t)$ lies on the circle $x^2 + y^2 = 1$ . That is where the name comes from.

Example At $x = \ln 2$ the first identity reads

\cosh^2 x - \sinh^2 x = \frac{25}{16} - \frac{9}{16} = 1.

For the second, $2x = \ln 4$ and

\sinh(\ln 4) = \frac{4 - 1/4}{2} = \frac{15}{8} = 2 \cdot \frac{3}{4} \cdot \frac{5}{4}.

Hyperbolic tangent

Definition For every real $x$ the hyperbolic tangent is

\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}.

It is odd, differentiable on $\mathbb{R}$ , has range $(-1, 1)$ , and

\frac{d}{dx} \tanh x = \frac{\cosh^2 x - \sinh^2 x}{\cosh^2 x} = \operatorname{sech}^2 x.

In plain words The graph of $\tanh$ is an S shape that rises from $-1$ to $1$ and has horizontal asymptotes $y = \pm 1$ .

Example At $\ln 2$ the value is $\tanh(\ln 2) = \frac{3/4}{5/4} = \frac{3}{5}$ , and the slope there is

1 - \tanh^2(\ln 2) = 1 - \frac{9}{25} = \frac{16}{25}.

Reciprocal hyperbolic functions

Definition The other three hyperbolic functions and their derivatives are

\operatorname{sech} x = \frac{1}{\cosh x}, \quad x \in \mathbb{R}, \qquad (\operatorname{sech} x)' = -\operatorname{sech} x \tanh x,

\operatorname{csch} x = \frac{1}{\sinh x}, \quad x \neq 0, \qquad (\operatorname{csch} x)' = -\operatorname{csch} x \coth x,

\coth x = \frac{\cosh x}{\sinh x}, \quad x \neq 0, \qquad (\coth x)' = -\operatorname{csch}^2 x.

In plain words These follow from the quotient rule. Notice that all three carry a minus sign, while among the trigonometric functions $\sec$ had none.

Example At $x = \ln 2$ we have $\coth x = \frac{5}{3}$ and $\operatorname{csch} x = \frac{4}{3}$ , so $(\coth x)' = -\frac{16}{9}$ .

Catenary

Definition A catenary is a curve of the form

y = a \cosh\left(\frac{x}{a}\right), \qquad a > 0,

up to shifting. It is the shape of a uniform, flexible chain hanging under its own weight. Its slope is $y' = \sinh\left(\frac{x}{a}\right)$ , and it satisfies the differential equation

y'' = \frac{1}{a} \sqrt{1 + (y')^2}.

In plain words The parameter $a$ sets how tight the chain is. Large $a$ gives a flat, shallow sag and small $a$ a deep one.

Example For $a = 2$ the curve is $y = 2 \cosh\left(\frac{x}{2}\right)$ . At $x = 2$ the height is $2 \cosh 1 \approx 3.086$ and the slope is $\sinh 1 \approx 1.175$ . The second derivative there is $\frac{1}{2} \cosh 1$ , which matches $\frac{1}{2}\sqrt{1 + \sinh^2 1}$ by the identity $\cosh^2 - \sinh^2 = 1$ .

Inverse hyperbolic functions

Definition The inverse hyperbolic functions have logarithm formulas,

\operatorname{arsinh} x = \ln\left(x + \sqrt{x^2 + 1}\right), \quad x \in \mathbb{R},

\operatorname{arcosh} x = \ln\left(x + \sqrt{x^2 - 1}\right), \quad x \ge 1,

\operatorname{artanh} x = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right), \quad -1 < x < 1.

Since $\cosh$ is not one-to-one, $\operatorname{arcosh}$ is the inverse of $\cosh$ restricted to $[0, \infty)$ , so its values are in $[0, \infty)$ . The reciprocal functions have inverses too, $\operatorname{arcoth} x = \operatorname{artanh}\frac{1}{x}$ for $\lvert x \rvert > 1$ , $\operatorname{arsech} x = \operatorname{arcosh}\frac{1}{x}$ for $0 < x \le 1$ , and $\operatorname{arcsch} x = \operatorname{arsinh}\frac{1}{x}$ for $x \neq 0$ . These are also written $\sinh^{-1} x$ , $\cosh^{-1} x$ and $\tanh^{-1} x$ . The formula for $\operatorname{arsinh}$ comes from solving

x = \frac{e^y - e^{-y}}{2},

which becomes the quadratic $(e^y)^2 - 2x e^y - 1 = 0$ in $e^y$ , and keeping the positive root $e^y = x + \sqrt{x^2 + 1}$ .

In plain words Because the hyperbolic functions are built from $e^x$ , their inverses are built from $\ln$ .

Example Since $\sqrt{(3/4)^2 + 1} = \frac{5}{4}$ , the formula gives

\operatorname{arsinh}\left(\frac{3}{4}\right) = \ln\left(\frac{3}{4} + \frac{5}{4}\right) = \ln 2,

which matches $\sinh(\ln 2) = \frac{3}{4}$ . Similarly $\sqrt{(5/4)^2 - 1} = \frac{3}{4}$ , so

\operatorname{arcosh}\left(\frac{5}{4}\right) = \ln\left(\frac{5}{4} + \frac{3}{4}\right) = \ln 2.

Derivative formulas and real domains

Definition Each inverse is differentiable on the open set shown.

Function	Derivative	Valid for
$\operatorname{arsinh} x$	$\frac{1}{\sqrt{x^2 + 1}}$	all real $x$
$\operatorname{arcosh} x$	$\frac{1}{\sqrt{x^2 - 1}}$	$x > 1$
$\operatorname{artanh} x$	$\frac{1}{1 - x^2}$	$\lvert x \rvert < 1$
$\operatorname{arcoth} x$	$\frac{1}{1 - x^2}$	$\lvert x \rvert > 1$
$\operatorname{arsech} x$	$-\frac{1}{x\sqrt{1 - x^2}}$	$0 < x < 1$
$\operatorname{arcsch} x$	$-\frac{1}{\lvert x \rvert \sqrt{1 + x^2}}$	$x \neq 0$

The function $\operatorname{arcosh}$ is defined at $x = 1$ but not differentiable there, because $\sinh 0 = 0$ and the graph has a vertical tangent.

In plain words Each formula comes from the inverse function rule plus an identity. The domain column matters, since $\operatorname{artanh}$ and $\operatorname{arcoth}$ share a formula but live on different sets.

Example By the table, the slope of $\operatorname{arsinh}$ at $\frac{3}{4}$ is

\frac{1}{\sqrt{9/16 + 1}} = \frac{1}{5/4} = \frac{4}{5}.

The inverse function rule gives the same, since $\operatorname{arsinh}\frac{3}{4} = \ln 2$ and $1 / \cosh(\ln 2) = \frac{4}{5}$ .

Check yourself

Differentiate $y = \sinh(x^2)$ .

Show the answer

By the chain rule $y' = 2x \cosh(x^2)$ .

Show that $\frac{d}{dx} \ln(\cosh x) = \tanh x$ and say where this holds.

Show the answer

By the chain rule

\frac{d}{dx} \ln(\cosh x) = \frac{\sinh x}{\cosh x} = \tanh x.

This holds for every real $x$ , since $\cosh x \ge 1 > 0$ .

Find the slope of $y = \operatorname{artanh} x$ at $x = \frac{1}{2}$ .

Show the answer

The derivative of $\operatorname{artanh} x$ is $\frac{1}{1 - x^2}$ for $\lvert x \rvert < 1$ , so the slope is

\frac{1}{1 - 1/4} = \frac{4}{3}.

Derivatives of Hyperbolic Functions

Theory, part 63

Practice, part 64

Key terms

Exponential definitions

Hyperbolic sine

Hyperbolic cosine

Identities

Hyperbolic tangent

Reciprocal hyperbolic functions

Catenary

Inverse hyperbolic functions

Derivative formulas and real domains

Check yourself

How this lesson connects

Builds on

Used later in