1. **Problem statement:** Find the differential uncertainty (doubt) $\Delta y$ for the function $$y = \left(\frac{1}{a} + \frac{1}{\sqrt{b}}\right) \cdot \frac{2}{c + d^2}$$ 2. **Step 1: Take the natural logarithm of both sides** to simplify differentiation using logarithmic properties: $$\ln(y) = \ln\left(\frac{1}{a} + b^{-1/2}\right) + \ln(2) - \ln(c + d^2)$$ 3. **Step 2: Differentiate both sides implicitly:** $$\frac{dy}{y} = \frac{-a^{-2} da - \frac{1}{2} b^{-3/2} db}{\frac{1}{a} + \frac{1}{\sqrt{b}}} - \frac{dc + 2 d \, dd}{c + d^2}$$ 4. **Step 3: Replace differentials $d$ with uncertainties $\Delta$ to express the relative uncertainty:** $$\frac{\Delta y}{y} = \frac{\frac{\Delta a}{a^2} + \frac{\Delta b}{2 b \sqrt{b}}}{\frac{1}{a} + \frac{1}{\sqrt{b}}} + \frac{\Delta c + 2 d \Delta d}{c + d^2}$$ 5. **Final formula for the uncertainty in $y$ is:** $$\boxed{\Delta y = y \times \left[ \frac{\frac{\Delta a}{a^2} + \frac{\Delta b}{2 b \sqrt{b}}}{\frac{1}{a} + \frac{1}{\sqrt{b}}} + \frac{\Delta c + 2 d \Delta d}{c + d^2} \right]}$$ This formula allows you to calculate the uncertainty in $y$ based on the uncertainties in $a$, $b$, $c$, and $d$.