How does $\frac{-e^{-x}}{\sqrt{e^{-2x}-1}}$ rearrange?

Question

I have been revising for my engineering mathematics exam which has a multiple choice question in it, which asks the following:

The derivative of $\arcsin(e^{-x})$ equals:

With several possible answers.

I have managed to find the derivative as far as $-\dfrac{e^{-x}}{\sqrt{e^{-2x}-1}}$ but this is not a valid option in the paper. I consulted an online calculator which rearranged the answer to $-\dfrac{1}{\sqrt{{e}^{2x}-1}}$ which is a valid option but the calculator doesn't explain the steps and I'd like to understand how to rearrange my answer to this form.

Thanks in advance.

Answer 1

$$-\frac{e^{-x}}{\sqrt{e^{-2x}-1}} = -\frac{1}{e^x\sqrt{e^{-2x}-1}}$$

$$-\frac{1}{\sqrt{e^{2x}}\sqrt{e^{-2x}-1}}$$