Author Topic: Should we know derivatives of hyperbolic functions?  (Read 1265 times)

Emily Deibert

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Should we know derivatives of hyperbolic functions?
« on: October 06, 2014, 11:15:20 PM »
Hi,

I've noticed in quite a few of the homework problems that we need to work with hyperbolic functions. I have never discussed hyperbolic functions in any math courses at U of T before (I've taken MAT135, MAT136, MAT235, and MAT223 prior to this course). Should I memorize the derivatives and properties of these functions for the test?

Thank you.

Victor Ivrii

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Re: Should we know derivatives of hyperbolic functions?
« Reply #1 on: October 07, 2014, 01:38:15 AM »
Remember that $\cosh(x)=\frac{1}{2}(e^x+e^{-x})$, $\sinh(x)=\frac{1}{2}(e^x-e^{-x})$ and therefore $(\cosh (x))'=\sinh(x)$, $(\sinh(x))'=\cosh(x)$ like fore ordinary $\cos(x)$ and $\sin(x)$ albeit without any minus sign, Further $\cosh^2(x)-\sinh^2(x)=1$.

Then $(\tanh(x))'=\cosh^{-2}(x)$, $(\coth(x))'=\cosh^{-2}(x)$ and so on.

Inverse hyperbolic functions could be expressed through logarithms http://en.wikipedia.org/wiki/Inverse_hyperbolic_function but to calculate their derivatives one needs just to remember derivatives of "direct" hyperbolic functions and how to differentiate inverse function. Then $(\operatorname{arcosh}(x))'=(x^2-1)^{-1/2}$, $(\operatorname{arsinh}(x))'=(x^2+1)^{-1/2}$, $(\operatorname{artanh}(x))'=(1-x^2)^{-1}$ ($|x|<1$), $(\operatorname{arcoth}(x))'=(1-x^2)^{-1}$ ($|x|>1$).

Even this is more than enough.