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

[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??

 

[Erland]

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??

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...

 

[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,
$\frac{d}{dx} (\tanh^{-1} \sinh 2x) = \frac{2 \cosh 2x}{1 - \sinh^2 2x}$