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If you hang a heavy electrical cable between two utility poles, it sags into a graceful curve called a catenary, modeled by the hyperbolic cosine function cosh⁡x\cosh x.
In the 1760s, Swiss mathematician Johann Heinrich Lambert introduced hyperbolic functions while studying non-Euclidean geometry and noticed that their calculus mirrored circular trigonometry, but with surprising sign changes.
Before finding their rates of change, what exact exponential building blocks make up these functions?

Exponential Foundations

The hyperbolic sine, sinh⁡x\sinh x, and hyperbolic cosine, cosh⁡x\cosh x, are defined using symmetric combinations of exponential growth exe^x and decay e−xe^{-x}.
What happens when we take the derivative of sinh⁡x\sinh x directly from this exponential definition?

Differentiating sinh(x)

Since the derivative of exe^x is exe^x and the derivative of e−xe^{-x} is −e−x-e^{-x}, differentiating sinh⁡x\sinh x is straightforward linear differentiation.
ddx(sinh⁡x)=ddx(ex−e−x2)=ex−(−e−x)2=ex+e−x2=cosh⁡x\frac{d}{dx}(\sinh x) = \frac{d}{dx}\left(\frac{e^x - e^{-x}}{2}\right) = \frac{e^x - (-e^{-x})}{2} = \frac{e^x + e^{-x}}{2} = \cosh x
Does differentiating cosh⁡x\cosh x introduce a negative sign the way circular cos⁡x\cos x does?

Differentiating cosh(x)