Question

Fill in the blanks to differentiate each of the given basic functions. You may assume that k, m, and b are appropriate constants:

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

Short answer

The solution is$\frac{d}{dx}\left(\sin hx\right)=\cos hx.$

Step-by-step solution

Step 1. Given information

The given function is$\frac{d}{dx}\left(\sin hx\right).$

Step 2. Calculation

We have,

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

$\sin hx=\frac{e^x-e^{-x}}{2}$

Taking derivative on both sides.

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

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