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

> The hyperbolic functions sinh x, cosh x and tanh x: exponential definitions, graphs, identities, derivatives, integrals and worked examples.

Source: https://practicecalculus.com/functions/hyperbolic-functions
Updated: 2026-11-03

$$ \sinh x = \frac{e^x - e^{-x}}{2}, \quad \cosh x = \frac{e^x + e^{-x}}{2}, \quad \tanh x = \frac{\sinh x}{\cosh x} $$

## At a glance

- **Domain of sinh x:** $(-\infty, \infty)$
- **Range of sinh x:** $(-\infty, \infty)$
- **Derivative:** $\cosh x$
- **Integral:** $\cosh x + C$
- **Key identity:** $\cosh^2 x - \sinh^2 x = 1$
- **Range of cosh x and tanh x:** $[1, \infty)$ and $(-1, 1)$

## Key takeaways

- $\sinh x$ and $\cosh x$ are built from $e^x$ and $e^{-x}$: they are its odd and even parts.
- Derivatives: $(\sinh x)' = \cosh x$, $(\cosh x)' = \sinh x$ and $(\tanh x)' = \operatorname{sech}^2 x$, with no minus signs.
- The main identity is $\cosh^2 x - \sinh^2 x = 1$, a minus where trig has a plus.
- Hyperbolic functions are not periodic: they grow exponentially (sinh, cosh) or level off (tanh).

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

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

Adding the first two gives $\cosh x + \sinh x = e^x$, so they split the [natural exponential function](/functions/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

```graph
title: 'Graphs of y = sinh x, y = cosh x and y = tanh x'
caption: '$\cosh x$ is a U-shaped curve with minimum $1$ at $x = 0$. $\sinh x$ passes through the origin and grows without bound. $\tanh x$ is squeezed between $-1$ and $1$.'
x: [-3, 3]
y: [-4, 4]
functions:
  - expr: sinh(x)
    label: y = sinh x
  - expr: cosh(x)
    label: y = cosh x
  - expr: tanh(x)
    label: y = tanh x
points:
  - [0, 1, '(0, 1)']
  - [0, 0, '(0, 0)']
```

| Function | Domain | Range | Symmetry | Value at $0$ |
| --- | --- | --- | --- | --- |
| $\sinh x$ | all reals | all reals | odd | $0$ |
| $\cosh x$ | all reals | $[1, \infty)$ | even | $1$ |
| $\tanh x$ | all reals | $(-1, 1)$ | odd | $0$ |

$\tanh x$ 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:

$$\cosh^2 x - \sinh^2 x = \frac{(e^x + e^{-x})^2 - (e^x - e^{-x})^2}{4} = \frac{4}{4} = 1$$

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

| Function | Derivative |
| --- | --- |
| $\sinh x$ | $\cosh x$ |
| $\cosh x$ | $\sinh x$ |
| $\tanh x$ | $\operatorname{sech}^2 x$ |

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

$$\frac{d}{dx}\sinh x = \frac{e^x - (-e^{-x})}{2} = \frac{e^x + e^{-x}}{2} = \cosh x$$

For $\tanh x$, the [quotient rule](/rules/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

| Function | Integral |
| --- | --- |
| $\sinh x$ | $\cosh x + C$ |
| $\cosh x$ | $\sinh x + C$ |
| $\tanh x$ | $\ln(\cosh x) + C$ |
| $\operatorname{sech}^2 x$ | $\tanh x + C$ |

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 $e^{-\ln 2} = \frac{1}{2}$,

$$\cosh(\ln 2) = \frac{2 + \frac{1}{2}}{2} = \frac{5}{4}$$

**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](/blog/trig-substitution) for integrals with $\sqrt{1 + x^2}$. Hyperbolic functions also have tidy [Taylor series](/blog/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](/blog/derivatives-of-exponential-and-logarithmic-functions).

## Frequently asked questions

### What is the derivative of sinh x and cosh x?

The derivative of $\sinh x$ is $\cosh x$ and the derivative of $\cosh x$ is $\sinh x$. Unlike $\cos x$, there is no minus sign.

### What is the derivative of tanh x?

The derivative of $\tanh x$ is $\operatorname{sech}^2 x = \frac{1}{\cosh^2 x}$, which also equals $1 - \tanh^2 x$.

### Why are they called hyperbolic functions?

The point $(\cosh t, \sinh t)$ always lies on the hyperbola $x^2 - y^2 = 1$, just as $(\cos t, \sin t)$ lies on the circle $x^2 + y^2 = 1$.

### What is a catenary?

A catenary is the curve a hanging chain or cable makes under its own weight. Its equation is $y = a\cosh\frac{x}{a}$.

### How do you pronounce sinh and cosh?

Common pronunciations are "sinch" or "shine" for $\sinh$, "kosh" for $\cosh$ and "tanch" for $\tanh$. Saying "hyperbolic sine" is always clear.