Derivatives of Hyperbolic Functions

The hyperbolic functions are simply a few handy combinations of $e^x$ and $e^{-x}$ that show up often enough to get their own names. ("Hyperbolic" because they trace out a hyperbola the way sine and cosine trace out a circle; more on that below.) The most famous one, $\cosh x$, is the exact shape of a cable or chain hanging between two posts. The definitions look intimidating, but each is a simple mix: $\cosh x$ (the average of $e^x$ and $e^{-x}$), $\sinh x$ (half their difference) and $\tanh x$ (one divided by the other, like $\tan$). Nothing new is going on here; it's all exponentials underneath.

The derivatives come straight from what you already know about exponentials: $e^x$ is its own derivative, and $e^{-x}$ has derivative $-e^{-x}$. So differentiating $\sinh x$ flips the sign on the $e^{-x}$ term:

which is $\cosh x$. Do the same to $\cosh x$ and you get $\sinh x$. Compare that with trig, where $(\cos x)' = -\sin x$. Here there's no minus sign.

The name. As $t$ varies, $(\cos t, \sin t)$ traces the circle $x^2 + y^2 = 1$. In the same way, $(\cosh t, \sinh t)$ traces the right half of the hyperbola $x^2 - y^2 = 1$. That's the identity $\cosh^2 t - \sinh^2 t = 1$, the hyperbolic cousin of $\cos^2 t + \sin^2 t = 1$ with one sign changed.

Shapes. $\cosh x$ is an even U-shaped curve with its minimum, 1, at $x = 0$. It looks like a parabola but rises faster, since it's made of exponentials. $\sinh x$ is odd and always increasing. $\tanh x$ is an S-curve squeezed between $-1$ and $1$.

Hanging cables and chains take the shape of a catenary, $y = a\cosh\frac xa$. The tanh S-curve shows up as a neural network activation and in the speed of an object falling with air resistance.