Differentials — Question 8

Question 8 – Solution

Strategy. Divide the total differential by $Q$, or equivalently differentiate the logarithmic power structure.

Step 1: Relative differential $$\ln Q=3\ln x+\frac 12\ln y-2\ln z.$$ Differentiating gives $$\boxed{\frac{dQ}{Q}=3\frac{dx}{x}+\frac 12\frac{dy}{y}-2\frac{dz}{z}}.$$

Step 2: Percentage estimate $$\frac{dQ}{Q}\approx 3(0.01)+\frac 12(-0.04)-2(0.005) =0.03-0.02-0.01=\boxed 0.$$ Thus the predicted first-order percentage change is $\boxed{0\%}$.

Step 3: Contributions The $x$ increase contributes $+3\%$, the $y$ decrease $-2\%$, and the $z$ increase $-1\%$. The $x$ term is individually largest, but the three first-order effects cancel exactly.