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

Adding and subtracting proof

if $y=x*z$ then ${\delta }_y=\sqrt{(z{\delta }_x{)}^2+(x{\delta }_z{)}^2}$

if $y=x/z$ then ${\delta }_y=\sqrt{{\left(\frac{{\delta }_x}{z}\right)}^2+{\left(\frac{x{\delta }_z}{z^2}\right)}^2}$

Algebra Proof

can not come up with one

Calculus Proof

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

$y=x/z$
then ${\delta }_y=\sqrt{{\left({\delta }_x\frac{\partial \left(\frac{x}{z}\right)}{\partial x}\right)}^2+{\left({\delta }_z\frac{\partial (\frac{x}{z})}{\partial z}\right)}^2}=\sqrt{{\left(\frac{{\delta }_x}{z}\right)}^2+{\left(\frac{-x{\delta }_z}{z^2}\right)}^2}=\sqrt{{\left(\frac{{\delta }_x}{z}\right)}^2+{\left(\frac{x{\delta }_z}{z^2}\right)}^2}$