Question

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

$\frac{d}{dx}\left(\cos hx\right)=\sin hx$

Short answer

We proved the formula$\frac{d}{dx}\left(\cos hx\right)=\sin hx$.

Step-by-step solution

Step 1. Given Information 

We need to prove the hyperbolic function$\frac{d}{dx}\left(\cos hx\right)=\sin hx$.

Step 2. Proving the function

$$\begin{gathered} \frac{d}{dx}\left(\cos hx\right) \\ =\frac{d}{dx}\left(\frac{e^x+e^{-x}}{2}\right) \\ =\frac{1}{2}\frac{d}{dx}\left(e^x\right)+\frac{d}{dx}\left(e^{-x}\right) \\ =\frac{1}{2}\left(e^x-e^{-x}\right) \\ =\sin hx \end{gathered}$$