Evaluate the derivative of the following functions.
f(x) = sin^-1 2x

Verified step by step guidance

1. Step 1: Recognize that \( f(x) = \sin^{-1}(2x) \) is the inverse sine function, also known as arcsine, applied to \( 2x \).
2. Step 2: Recall the derivative formula for the inverse sine function: \( \frac{d}{dx}[\sin^{-1}(u)] = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \).
3. Step 3: Identify \( u = 2x \) in this problem, so you need to find \( \frac{du}{dx} \).
4. Step 4: Differentiate \( u = 2x \) with respect to \( x \), which gives \( \frac{du}{dx} = 2 \).
5. Step 5: Substitute \( u = 2x \) and \( \frac{du}{dx} = 2 \) into the derivative formula: \( \frac{d}{dx}[\sin^{-1}(2x)] = \frac{1}{\sqrt{1-(2x)^2}} \cdot 2 \).