Question 2.9.2: Finding a Formula for an Inverse Hyperbolic Function Find an...

Recall from (9.2) that

y=\sinh ^{-1} x \quad \text { if and only if } \sinh y=x (9.2)

y=\sinh ^{-1} x \quad \text { if and only if } \sinh y=x \text {. }

Using this definition, we have

x=\sinh y=\frac{e^y-e^{-y}}{2}. (9.3)

We can solve this equation for y, as follows. First, recall also that

\cosh y=\frac{e^y+e^{-y}}{2}.

Now, notice that adding these last two equations and using the identity (9.1), we have

\rho\left(x_1\right)=\lim _{x \rightarrow x_1} \frac{f(x)-f\left(x_1\right)}{x-x_1}=f^{\prime}\left(x_1\right)     (9.1)

e^y=\sinh y+\cosh y=\sinh y+\sqrt{\cosh ^2 y} Since cosh y > 0

=x+\sqrt{x^2+1} \text {, }

from (9.3). Finally, taking the natural logarithm of both sides, we get

That is, we have found a formula for the inverse hyperbolic sine function:

\sinh ^{-1} x=\ln \left(x+\sqrt{x^2+1}\right).