I am trying to prove that the uncertainty of the equation $$Q = xy$$ is equal to $$\frac{\Delta Q}{Q_0} = \frac{\Delta x}{x} + \frac{\Delta y}{y}.$$
However what I am getting is $$\Delta Q =\sqrt{(y\Delta X)^ 2 + (x \Delta y)^²} $$ and I am stuck there.. How to continue to prove that it is equal to the second equation that I have written?
$$\left ( \frac{\Delta Q}{Q_0} \right)^2 = \left ( \frac{\Delta x}{x} \right )^2 + \left ( \frac{\Delta y}{y} \right )^2. $$
In your last step, divide by $Q_0 = xy$.
If I understand correctly you are trying to derive
$$\frac{\Delta Q}{Q_0} = \frac{\Delta x}{x} + \frac{\Delta y}{y}.$$
for
for $$Q = xy$$
and you are using the propagation of error general formula in order to do that:
$$\Delta Q =\sqrt{( \frac {\partial Q}{\partial x}\Delta x)^2+(\frac {\partial Q}{\partial y}\Delta y)^2} $$
But you are getting
$$\Delta Q =\sqrt{(y\Delta X)^ 2 + (x \Delta y)^²} $$
I think two points are relevant here.
1) as @SuperCioca points out you should divide by Q to get $${\Delta Q \over Q}=\sqrt{({y\Delta X \over Q})^ 2 + ({x \Delta y\over Q})^²} $$ which simplifies to $${\Delta Q \over Q}=\sqrt{({\Delta X \over x})^ 2 + ({\Delta y\over y})^²} $$
2) It is impossible to get the error propagation expression
$$\frac{\Delta Q}{Q_0} = \frac{\Delta x}{x} + \frac{\Delta y}{y}.$$
from the general formula - The expression that you have been asked to derive is a simplification that is sometimes used because it is simpler that the `correct' version of that you have derived. This expression you are trying to derive is considered simpler because it does not involve squaring and taking the square root. From another point of view the expression that you are trying to derived gives the 'maximum' possible error given errors in x and y of $\Delta$x and $\Delta$y. It is easier to show that $\Delta$Q = $\Delta$x + $\Delta$y is the maximum possible error for Q=x+y -- although the more correct version for the error in Q=x+y is a $\sqrt{ (\Delta x^2 +\Delta y^2)}$.
So in essence the expression you want to derive is a 'maximum possible error', but the general formula give a 'statistically likely error'.
For maximum possible error...
if $$Q=xy $$ $$Q + \Delta Q= (x+\Delta x)(y+\Delta y)$$ $$Q + \Delta Q= xy +y\Delta x + x\Delta y + \Delta x\Delta y $$ now xy = Q so $$ \Delta Q= y\Delta x + x\Delta y + \Delta x\Delta y $$ Now $\Delta x\Delta y$ should be small enough to ignore.... and we divide by Q $$ {\Delta Q \over Q}= {y\Delta x\over Q} + {x\Delta y \over Q} $$ hence $$ {\Delta Q \over Q}= {\Delta x\over x} + {\Delta y \over y} $$
But as I hope is clear by now this formula is not as good as the one you derived from the general error formula
