General Engineering Introduction/Error Analysis/Calculus of Error/multiply proof

Multiplying by a Constant Proof

if $y=C\ast x$ then ${\delta }_y=C\ast {\delta }_x$

Algebra Proof

start with the formula: dependent equals constant times independent
$y=C\ast x$
substitute the measured value of x (and x's error) and set equal to y plus the unknown error in y that we are interested in:
$y+{\delta }_y=C(x+{\delta }_x)=C\ast x+C\ast {\delta }_x$
subtract the top equation from the bottom:
$\cancel{y}+{\delta }_y=\cancel{C\ast x}+C\ast {\delta }_x$
and are left with the equation: error in y equals constant times error in x
${\delta }_y=C\ast {\delta }_x$

Calculus Proof

start with the formula: dependent equals constant times independent
$y=C\ast x$
then ${\delta }_y=\sqrt{{\left({\delta }_x\frac{\partial (C\ast x)}{\partial x}\right)}^2}={\delta }_x\ast C\frac{\partial x}{\partial x}={\delta }_x\ast C=C\ast {\delta }_x$