Law of Propagation of Uncertainty

f(x,y,...)f(x,y,...)
δf(x,y,...)=(∂f∂xδx)2+(∂f∂yδy)2+...\delta f(x,y,...)=\sqrt{\left(\frac{\partial f}{\partial x} \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \delta y \right)^2 + ...}

Add/Subtract

f(x,y,...)=x+y+...f(x,y,...)=x+y+...
δf(x,y,...)=(∂f∂xδx)2+(∂f∂yδy)2+...=(δx)2+(δy)2+... \begin{aligned} \delta f(x,y,...) &= \sqrt{\left(\frac{\partial f}{\partial x} \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \delta y \right)^2 + ...} \\ &= \sqrt{\left(\delta x \right)^2 + \left(\delta y \right)^2 + ...} \\ \end{aligned}

Multiply/Divide

f(x,y,...)=x∗y∗...f(x,y,...)=x*y*...
δf(x,y,...)=(∂f∂xδx)2+(∂f∂yδy)2+...=((y∗...)δx)2+((x∗...)δy)2+...δf(x,y,...)(x∗y∗...)=1(x∗y∗...)((y∗...)δx)2+((x∗...)δy)2+...=(1(x∗y∗...))2((y∗...)δx)2+(1(x∗y∗...))2((x∗...)δy)2+...=(1x)2(δx)2+(1y)2(δy)2+...=(δxx)2+(δyy)2+... \begin{aligned} \delta f(x,y,...) &= \sqrt{\left(\frac{\partial f}{\partial x} \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \delta y \right)^2 + ...} \\ &= \sqrt{\left((y * ...) \delta x \right)^2 + \left((x * ...) \delta y \right)^2 + ...} \\ \frac{\delta f(x,y,...)}{(x * y * ...)} &= \frac{1}{(x * y * ...)} \sqrt{\left((y * ...) \delta x \right)^2 + \left((x * ...) \delta y \right)^2 + ...} \\ &= \sqrt{\left(\frac{1}{(x * y * ...)}\right)^2 \left((y * ...) \delta x \right)^2 + \left(\frac{1}{(x * y * ...)}\right)^2 \left((x * ...) \delta y \right)^2 + ...} \\ &= \sqrt{\left(\frac{1}{x}\right)^2 \left(\delta x \right)^2 + \left(\frac{1}{y}\right)^2 \left(\delta y \right)^2 + ...} \\ &= \sqrt{\left(\frac{\delta x}{x}\right)^2 + \left(\frac{\delta y}{y}\right)^2 + ...} \\ \end{aligned}

Exponent

f(x,y)=xyf(x,y)=x^y
δf(x,y)=(∂f∂xδx)2+(∂f∂yδy)2+...=((y∗xy−1)δx)2+((xyln(x))δy)2+...δf(x,y)xy=1xy((y∗xy−1)δx)2+((xyln(x))δy)2+...=(1xy)2((y∗xy−1)δx)2+(1xy)2((xyln(x))δy)2+...=(yxδx)2+(ln(x)δy)2+... \begin{aligned} \delta f(x,y) &= \sqrt{\left(\frac{\partial f}{\partial x} \delta x \right)^2 + \left(\frac{\partial f}{\partial y} \delta y \right)^2 + ...} \\ &= \sqrt{\left(\left(y * x ^{y - 1} \right) \delta x \right)^2 + \left(\left(x^y ln(x) \right) \delta y \right)^2 + ...} \\ \frac{\delta f(x,y)}{x^y} &= \frac{1}{x^y}\sqrt{\left(\left(y * x ^{y - 1} \right) \delta x \right)^2 + \left(\left(x^y ln(x) \right) \delta y \right)^2 + ...} \\ &= \sqrt{\left(\frac{1}{x^y}\right)^2\left(\left(y * x ^{y - 1} \right) \delta x \right)^2 + \left(\frac{1}{x^y}\right)^2 \left(\left(x^y ln(x) \right) \delta y \right)^2 + ...} \\ &= \sqrt{\left(\frac{y}{x} \delta x \right)^2 + \left(ln(x) \delta y \right)^2 + ...} \\ \end{aligned}