12.54 Find the function $S(x)$, if its derivative $S^{\prime}(x)=\frac{2}{\sqrt{5-x}}$ and $S(1)=-1$.

Calculate the integrals (12.55-12.58):
12.54 Let's write the derivative in the form \( S^{\prime}(x)=2(5-x)^{-1 / 2} \). Then, using rule \( 3^{0} \) v), we find

\[
S(x)=2 \frac{1}{-1} \frac{(5-x)^{1 / 2}}{\frac{1}{2}}+C=-4 \sqrt{5-x}+C
\]

Setting \( x=1 \), we get \( S(1)=-4 \cdot 2+C=-1 \), from which \( C=7 \). Therefore, \( S(x)=7-4 \sqrt{5-x} \).