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 $$\delta_y = \sqrt{\left ( \delta_x \frac{\partial {\left ( x*z \right )}} \right ) ^2 + \left ( \delta_z \frac{\partial (x*z)}{\partial z} \right ) ^2} = \sqrt{(z\delta_x)^2 + (x\delta_z)^2}$$

$$y = x/z$$ then $$\delta_y = \sqrt{\left ( \delta_x \frac{\partial {\left (\frac{x}{z} \right )}} \right ) ^2 + \left ( \delta_z \frac{\partial (\frac{x}{z})}{\partial z} \right ) ^2} = \sqrt{\left ( \frac{\delta_x}{z} \right ) ^2 + \left ( \frac{-x \delta_z}{z^2} \right ) ^2} = \sqrt{\left ( \frac{\delta_x}{z} \right ) ^2 + \left ( \frac{x \delta_z}{z^2} \right ) ^2}$$