A3. Which of the following functions is the derivative of the function $f(x)=\frac{1-2 x}{\sqrt{2 x}}$?
(A) $f^{\prime}(x)=\frac{-\sqrt{2 x}(2 x-1)}{4 x^{2}}$
(B) $f^{\prime}(x)=\frac{-\sqrt{2 x}(2 x-1)}{8 x^{2}}$
(C) $f^{\prime}(x)=\frac{-\sqrt{2 x}(2 x+1)}{4 x^{2}}$
(D) $f^{\prime}(x)=\frac{-\sqrt{2 x}(2 x+1)}{8 x^{2}}$
(E) $f^{\prime}(x)=\frac{-\sqrt{2 x}(1-2 x)}{4 x^{2}}$
A3. We calculate the derivative of the function $f$ according to the formula $f^{\prime}(x)=\frac{-2 \sqrt{2 x}-(1-2 x) \cdot \frac{1}{2}(2 x)^{-\frac{1}{2}} \cdot 2}{2 x}$. Expand the numerator to a common denominator, resolve the double fraction, and by rationalizing the fraction, we get $f^{\prime}(x)=$ $\frac{-2 x \sqrt{2 x}-\sqrt{2 x}}{4 x^{2}}$. Factor out the common factor in the numerator and obtain the result $f^{\prime}(x)=\frac{-\sqrt{2 x}(2 x+1)}{4 x^{2}}$.