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