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

Multiplying by a Constant Proof

if $y=C*x$ then ${\delta _{y}}=C*{\delta _{x}}$

Algebra Proof

start with the formula: dependent equals constant times independent
$y=C*x$
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:
$y+\delta _{y}=C(x+\delta _{x})=C*x+C*\delta _{x}$
subtract the top equation from the bottom:
${\cancel {y}}+\delta _{y}={\cancel {C*x}}+C*\delta _{x}$
and are left with the equation: error in y equals constant times error in x
$\delta _{y}=C*\delta _{x}$

Calculus Proof

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