Article 15
Derivatives of Anti-hyperbolic Functions.

(a)

Let $u={sinh}^{-1}x$, then $x=sinhu$, $dx=coshudu=\sqrt{1+{sinh}^{2}u}=\sqrt{1+x^2}du$, $du=\frac{dx}{\sqrt{1+x^2}}$.

(b)

Similar to (a).

(c)

Let $u={tanh}^{-1}x$, then $x=tanhu$, , $du=\frac{dx}{1-x^2}$.

(d)

Similar to (c).

(e)

(f)

Similar to (e).

Prob. 31.

Prove

Prob. 32.

Prove $$\begin{array}{llllllll}\hfill d{sinh}^{-1}\frac{x}{a}& =\frac{dx}{\sqrt{x^2+a^2}},& \hfill d{cosh}^{-1}\frac{x}{a}& =\frac{dx}{\sqrt{x^2-a^2}},& \hfill & & \hfill & \\ \hfill d{tanh}^{-1}\frac{x}{a}& ={\left.\frac{adx}{a^2-x^2}\right]}_{x<a},& \hfill d{coth}^{-1}\frac{x}{a}& =-{\left.\frac{adx}{x^2-a^2}\right]}_{x>a}.& \hfill & & \hfill & \end{array}$$

 

Prob. 33.

Find independently of ${cosh}^{-1}x$.

Prob. 34.

When ${tanh}^{-1}x$ is real, prove that ${coth}^{-1}x$ is imaginary, and conversely; except when $x=1$.

Prob. 35.

Evaluate $\frac{{sinh}^{-1}x}{logx}$, $\frac{{cosh}^{-1}x}{logx}$ when $x=\infty$.