Chapter 2: Differentiation
Section 2.10: The Inverse Hyperbolic Functions and Their Derivatives
Establish the identity arcsinhx=lnx+x2+1.
The identity arcsinhx=lnx+x2+1 can be established as follows.
Set y=arcsinhx so x=sinhy=ey−e−y2. Set u=ey so e−y=1/u and the equation x=u−1/u2 becomes the quadratic (in u)
u2−2 x u−1=0
having solution u=x ±x2+1. Because u is an exponential, it must be positive, so
from which it follows that y=arcsinhx=lnx+x2+1.
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