How to find derivative of tanh^-1(x)

1. Apr 11, 2014

sozener1

how do I find the derivative of tanh^-1(sinh(2x))

do I just find derivative of tanh^-1(x) this then substitute sinh(2x) into x??

2. Apr 11, 2014

Erland

Yes, but since the Chain Rule applies here, you must not forget the inner derivative.

It is also possible that this can be rewritten and simplified, but I'm too tired to figure this out now...

3. Apr 13, 2014

AMenendez

Chain Rule, Chain Rule, Chain Rule!
$\frac{d}{dx} (\tanh^{-1} \sinh 2x) = \frac{d}{dx} (\tanh^{-1} \sinh 2x) \cdot \frac{d}{dx} \sinh 2x = \frac{1}{1 - \sinh^2 2x} \cdot 2\cosh2x$
Thus,
$\frac{d}{dx} (\tanh^{-1} \sinh 2x) = \frac{2 \cosh 2x}{1 - \sinh^2 2x}$