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

if ${\displaystyle y=x*z}$ then ${\displaystyle \delta _{y}={\sqrt {(z\delta _{x})^{2}+(x\delta _{z})^{2}}}}$
if ${\displaystyle y=x/z}$ then ${\displaystyle \delta _{y}={\sqrt {\left({\frac {\delta _{x}}{z}}\right)^{2}+\left({\frac {x\delta _{z}}{z^{2}}}\right)^{2}}}}$

Algebra Proof

can not come up with one

Calculus Proof

${\displaystyle y=x*z}$
then ${\displaystyle \delta _{y}={\sqrt {\left(\delta _{x}{\frac {\partial {\left(x*z\right)}}{\partial x}}\right)^{2}+\left(\delta _{z}{\frac {\partial (x*z)}{\partial z}}\right)^{2}}}={\sqrt {(z\delta _{x})^{2}+(x\delta _{z})^{2}}}}$

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