Inverse Hyperbolic Trig Functions?

**elbarto** · September 10th 2007, 05:26 AM

Hi,
Can somebody please help me with finding the derivative of this function? I can find the derivative if it is a normal function but the inverse function has me stumped at the moment.

Thanks

Elbarto

**topsquark** · September 10th 2007, 05:50 AM

Originally Posted by elbarto

Hi,
Can somebody please help me with finding the derivative of this function? I can find the derivative if it is a normal function but the inverse function has me stumped at the moment.

Thanks

Elbarto

Are you looking for the derivative of $ar~sinh(2x)$ or the derivative of its inverse?
$y^{\prime} = ar~cosh(2x) \cdot 2 = 2ar~cosh(2x)$
by the chain rule
and note that $\frac{d}{dx}sinh^{-1}(x) = \frac{1}{\sqrt{x^2 + 1}}$
so
$y = \frac{1}{2}sinh^{-1} \left ( \frac{x}{ar} \right )$
is the inverse function and has a derivative
$y^{\prime} = \frac{1}{2} \cdot \frac{1}{\sqrt{ \left ( \frac{x}{ar} \right )^2 + 1}} \cdot \frac{1}{ar} = \frac{1}{2\sqrt{x^2 + a^2r^2}}$
after some simplification.

-Dan

**elbarto** · September 10th 2007, 06:08 AM

Hi dan,

Im just trying to find the derivative of y. is it as simple as
$y^{\prime} = ar~cosh(2x) \cdot 2 = 2ar~cosh(2x)$ ?

Thanks for your help

**Soroban** · September 10th 2007, 06:28 AM

Hello, Elbarto!

There is a formula for this problem.
If you don't know it, we can derive it.

$y \;=\;\sinh^{-1}(4x)$

Take the $\sinh$ of both sides: . $\sinh y \;=\;4x$ .[1]

Differentiate implicitly: . $\cosh y\cdot\frac{dy}{dx}\;=\;4\quad\Rightarrow\quad\fra c{dy}{dx}\;=\;\frac{4}{\cosh y}<br />$ .[2]

From the identity: . $\cosh^2u - \sinh^2u \:=\:1$ , we have: . $\cosh u \:=\:\sqrt{1 + \sinh^2u}$

Then [2] becomes: . $\frac{dy}{dx} \;=\;\frac{4}{\sqrt{1+\sinh^2y}}$

Substitute [1] and we have: . $\frac{dy}{dx}\;=\;\frac{4}{\sqrt{1 + (4x)^2}} \;=\;\frac{4}{\sqrt{1+16x^2}}$

**topsquark** · September 10th 2007, 10:24 AM

Oh! That "ar" on the outside of the sinh(x), that was supposed to mean "arcsinh(x)"? Sorry about that.

-Dan

Thread: Inverse Hyperbolic Trig Functions?

LinkBack

Thread Tools

Display

Inverse Hyperbolic Trig Functions?

Similar Math Help Forum Discussions

Integration of inverse hyperbolic functions

inverse hyperbolic functions and hyperbola area

inverse hyperbolic functions

Need Calculus Help - Hyperbolic Trig Functions

hyperbolic functions inverse

Search Tags