Answer
${dy}={\frac{-1}{|e^{(-x)}|{\sqrt{((e^{-x})^2-1})}}}\times{{(e^{(-x)})}}{dx}$
Work Step by Step
We are given: $y={\sec^{-1}{(e^{-x}})}$
on differentiating both sides:
${\frac{dy}{dx}}={\frac{d({\sec^{-1}{e^{-x}}})}{dx}}$
on applying the chain rule:
${\frac{dy}{dx}}={\frac{1}{|e^{(-x)}|{\sqrt{((e^{-x})^2-1})}}}\times{\frac{d(e^{(-x)})}{dx}}$
${\frac{dy}{dx}}={\frac{1}{|e^{(-x)}|{\sqrt{((e^{-x})^2-1})}}}\times{{-(e^{(-x)})}}$
${dy}={\frac{-1}{|e^{(-x)}|{\sqrt{((e^{-x})^2-1})}}}\times{{(e^{(-x)})}}{dx}$
The final answer is: ${dy}={\frac{-1}{|e^{(-x)}|{\sqrt{((e^{-x})^2-1})}}}\times{{(e^{(-x)})}}{dx}$