Question

Prove each of the differentiation formulas in Exercises 93–96. (These exercises involve hyperbolic functions.)

$\frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sqrt{x^2-1}}$.

Short answer

We proved the formula$\frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sqrt{x^2-1}}$.

Step-by-step solution

Step 1. Given Information 

We need to prove the hyperbolic function $\frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sqrt{x^2-1}}$.

Step 2. Proving the function  

$$\begin{gathered} \cos h\left(\cos h^{-1}x\right)=x \\ \Rightarrow \frac{d}{dx}\cos h\left(\cos h^{-1}x\right)=\frac{d}{dx}\left(x\right) \\ \Rightarrow \sin h\left(\cos h^{-1}x\right)\frac{d}{dx}\left(\cos h^{-1}x\right)=1 \\ \Rightarrow \frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sin h\left(\cos h^{-1}x\right)} \\ \Rightarrow \frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sqrt{\cos h^2\left(\cos h^{-1}x\right)-1}} \\ \Rightarrow \frac{d}{dx}\left(\cos h^{-1}x\right)=\frac{1}{\sqrt{x^2-1}} \end{gathered}$$