Derivative of cosh x: Formula, Proof and Examples

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

Proof from the definition

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

Differentiate each piece. The derivative of eˣ is e^x, and the derivative of e^(−x) is -e^{-x}:

Graph of cosh x and its derivative

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

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:

Chain rule version and examples

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:

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

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, the derivative of tanh x and the hyperbolic functions page for graphs and identities.