# Derivative of cosh x: Formula, Proof and Examples

> The derivative of cosh x is sinh x, with no minus sign. See the proof from exponentials, the hanging cable example, chain rule examples and a graph.

Source: https://practicecalculus.com/derivatives/cosh-x
Updated: 2026-11-06

$$ \frac{d}{dx}\,\cosh x = \sinh x $$

## At a glance

- **Derivative:** $\sinh x$
- **Definition:** $\cosh x = \dfrac{e^x + e^{-x}}{2}$
- **Chain rule form:** $\dfrac{d}{dx}\cosh u = \sinh u\cdot u'$
- **Key identity:** $\cosh^2 x - \sinh^2 x = 1$
- **Inverse derivative:** $\dfrac{d}{dx}\operatorname{arcosh} x = \dfrac{1}{\sqrt{x^2 - 1}}$ for $x > 1$

## Key takeaways

- The derivative of $\cosh x$ is $\sinh x$, with no minus sign.
- The proof uses $\cosh x = \frac{e^x + e^{-x}}{2}$ and the derivative of $e^{-x}$.
- Since $\sinh x$ changes sign at $0$, $\cosh x$ falls for $x < 0$, rises for $x > 0$, and has a minimum of $1$ at $x = 0$.
- With the chain rule, $\frac{d}{dx}\cosh u = \sinh u\cdot u'$.

The **derivative of $\cosh x$ is $\sinh x$**. Watch the sign: unlike the circular cosine, there is no minus.

$$\frac{d}{dx}\,\cosh x = \sinh x$$

## Proof from the definition

The hyperbolic cosine is the average of $e^x$ and $e^{-x}$:

$$\cosh x = \frac{e^x + e^{-x}}{2}$$

Differentiate each piece. The [derivative of eˣ](/derivatives/e-to-the-x) is $e^x$, and the [derivative of e^(−x)](/derivatives/e-to-the-negative-x) is $-e^{-x}$:

$$\frac{d}{dx}\,\frac{e^x + e^{-x}}{2} = \frac{e^x - e^{-x}}{2} = \sinh x$$

## Graph of cosh x and its derivative

```graph
title: 'Graph of y = cosh x and its derivative y = sinh x'
caption: 'The hyperbolic cosine has a U shape with its minimum at $(0, 1)$. Its slope, $\sinh x$, is negative on the left, zero at the bottom and positive on the right.'
x: [-3, 3]
y: [-6, 6]
functions:
  - expr: cosh(x)
    label: y = cosh x
  - expr: sinh(x)
    label: y = sinh x (derivative)
    dashed: true
points:
  - [0, 1, 'minimum (0, 1)']
  - [0, 0, 'slope 0']
```

The derivative $\sinh x$ is negative for $x < 0$ and positive for $x > 0$, so $x = 0$ is a minimum. The second derivative is $\cosh x \ge 1 > 0$, so the curve is concave up everywhere.

## The hanging cable

A chain or cable hanging under its own weight takes the shape $y = a\cosh\frac{x}{a}$, called a catenary. Its slope at any point is

$$\frac{dy}{dx} = a\sinh\frac{x}{a}\cdot\frac{1}{a} = \sinh\frac{x}{a}$$

So the cable is flat at its lowest point and gets steeper toward the supports. The arc length formula also simplifies nicely, because $1 + \sinh^2 u = \cosh^2 u$:

$$\sqrt{1 + \left(\frac{dy}{dx}\right)^2} = \sqrt{1 + \sinh^2\frac{x}{a}} = \cosh\frac{x}{a}$$

## Chain rule version and examples

$$\frac{d}{dx}\cosh u = \sinh u\cdot u'$$

| Function | Derivative |
| --- | --- |
| $\cosh(2x)$ | $2\sinh(2x)$ |
| $\cosh(x^2)$ | $2x\sinh(x^2)$ |
| $\cosh(\ln x)$ | $\frac{\sinh(\ln x)}{x} = \frac{1}{2}\left(1 - \frac{1}{x^2}\right)$ |
| $\cosh^2 x$ | $2\cosh x\sinh x = \sinh 2x$ |
| $x\cosh x$ | $\cosh x + x\sinh x$ |

In the third row, $\sinh(\ln x) = \frac{x - 1/x}{2}$, so dividing by $x$ gives $\frac{1}{2}\left(1 - \frac{1}{x^2}\right)$.

**Worked example: checking the identity.** Differentiate $\cosh^2 x - \sinh^2 x$:

$$2\cosh x\sinh x - 2\sinh x\cosh x = 0$$

A zero derivative means the expression is constant. At $x = 0$ it equals $1 - 0 = 1$, which proves $\cosh^2 x - \sinh^2 x = 1$.

## Derivative of arcosh x

For $x > 1$, the inverse hyperbolic cosine has derivative

$$\frac{d}{dx}\operatorname{arcosh} x = \frac{1}{\sqrt{x^2 - 1}}$$

Set $\cosh y = x$ with $y > 0$, so $\sinh y\,y' = 1$ and $\sinh y = \sqrt{\cosh^2 y - 1} = \sqrt{x^2 - 1}$.

## Common mistakes

1. **Writing $-\sinh x$.** Copying the circular pattern is the most common error.
2. **Thinking $\cosh x$ can be below 1.** Its minimum is $\cosh 0 = 1$.
3. **Forgetting the inside derivative.** $\frac{d}{dx}\cosh(4x) = 4\sinh(4x)$.

See the [derivative of sinh x](/derivatives/sinh-x), the [derivative of tanh x](/derivatives/tanh-x) and the [hyperbolic functions](/functions/hyperbolic-functions) page for graphs and identities.

## Frequently asked questions

### What is the derivative of cosh x?

The derivative of $\cosh x$ is $\sinh x$.

### Is the derivative of cosh x equal to −sinh x?

No. That is the pattern for circular cosine, $\frac{d}{dx}\cos x = -\sin x$. For the hyperbolic cosine the answer is $+\sinh x$.

### What is the derivative of cosh(2x)?

By the chain rule it is $2\sinh(2x)$.

### What is the second derivative of cosh x?

It is $\cosh x$ again. Differentiating twice brings you back to the start.