Chapter 4.3, Problem 37E

To determine

To find: The correct option.

Answer to Problem 37E

The correct option is E .

Explanation of Solution

Given Information: The function is $\frac{d}{dx}{\sin }^{-1}\left(\frac{x}{2}\right)$ .

Calculation:

The derivative of the function is,

  $\begin{array}{c}\frac{d}{dx}{\sin }^{-1}\left(\frac{x}{2}\right)=\frac{1}{\sqrt{1-{\left(\frac{x}{2}\right)}^2}}\times \frac{d}{dx}\left(\frac{x}{2}\right)\left(\because \frac{d}{dx}{\sin }^{-1}x=\frac{1}{\sqrt{1-x^2}}\right)\\ =\frac{1}{\sqrt{1-\frac{x^2}{4}}}\times \left(\frac{1}{2}\right)\\ =\frac{1}{2\sqrt{\frac{4-x^2}{4}}}\\ =\frac{2}{2\sqrt{4-x^2}}\\ =\frac{1}{\sqrt{4-x^2}}\end{array}$

Conclusion:

The correct option is E .