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

Verified step by step guidance

1. Step 1: Recognize that the function f(x) = \sin^{-1}(e^{-2x}) is an inverse trigonometric function composed with an exponential function. To find the derivative, we will use the chain rule.
2. Step 2: Recall the derivative of the inverse sine function: \( \frac{d}{dx} [\sin^{-1}(u)] = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \). Here, u = e^{-2x}.
3. Step 3: Differentiate the inner function u = e^{-2x} with respect to x. The derivative of e^{-2x} is \( \frac{d}{dx}[e^{-2x}] = -2e^{-2x} \).
4. Step 4: Substitute the derivative of the inner function and the expression for u into the derivative formula for the inverse sine function: \( \frac{d}{dx} [\sin^{-1}(e^{-2x})] = \frac{1}{\sqrt{1-(e^{-2x})^2}} \cdot (-2e^{-2x}) \).
5. Step 5: Simplify the expression: The derivative is \( \frac{-2e^{-2x}}{\sqrt{1-e^{-4x}}} \).