6.12. Find the derivative of the complex three-link function

$$
f(x)=\arcsin \left(\sqrt{1-e^{x}}\right), x \leqslant 0
$$
Solution.

Here, the first outer link is the arcsine, the second is the square root, and the third is the sum of one and the exponent. We differentiate sequentially:

$$
\begin{aligned}
& f^{\prime}(x)=\frac{1}{\sqrt{1-\left(\sqrt{1-e^{x}}\right)^{2}}} \cdot\left(\sqrt{1-e^{x}}\right)^{\prime}=\frac{1}{\sqrt{1-\left(1-e^{x}\right)}} \cdot \frac{1}{2 \sqrt{1-e^{x}}} \cdot\left(1-e^{x}\right)^{\prime}= \\
& =\frac{1}{\sqrt{1-\left(1-e^{x}\right)}} \cdot \frac{1}{2 \sqrt{1-e^{x}}} \cdot\left(0-e^{x}\right)=\frac{-e^{x}}{2 \cdot e^{x / 2} \sqrt{1-e^{x}}}=\frac{-e^{x / 2}}{2 \sqrt{1-e^{x}}}, x<0 .
\end{aligned}
$$