# Trivial derivations about error analysis (combination of errors)

My book deals with the combination of errors for addition, multiplication and for exponents. I have understood the derivation of each but I am struggling to extend the same for division and negative powers.

For Errors of Multiplication : Let $$Z$$ = $$AB$$, thus $$Z \pm \delta Z$$ = $$(A\pm\delta A)(B\pm \delta B)$$

i.e : $$Z \pm \delta Z$$ = $$AB\pm B\delta A\pm A\delta B\pm \delta A\delta B$$

Dividing $$LHS$$ by $$Z$$ and $$RHS$$ by $$AB$$ gives :

$$\frac{\delta Z}{Z}$$ = $$\frac {\delta A}{A}$$ $$\pm$$ $$\frac{\delta B}{B}$$

Question 1:

$$a)$$ For the derivation regarding division how do you proceed beyond $$Z \pm \delta Z$$ = $$\frac{(A\pm\delta A)}{(B\pm \delta B)}$$?

$$b)$$ If not via the above method , how is the result for division derived?

For Errors in Exponents : $$Z$$ = $$A^2$$ which gives : $$\frac{\delta Z}{Z}$$ = $$\frac{2\delta A}{A}$$

This is generalized to give : $$\frac{\delta Z}{Z}$$ = $$p \frac{\delta A}{A}$$+ $$q\frac{\delta B}{B}$$+$$r\frac{\delta C}{C}$$ where $$Z$$ = $$\frac{(A^p)(B^q)}{C^r}$$

Question 2:

$$c)$$ How is the result arrived at for negative powers?

• I forgot to precise that $\frac{\Delta X}X$ stands for $\left|\frac{\Delta X}{X}\right|$ Commented Apr 7, 2019 at 9:10

In my humble opinion, the most convenient way is to use logarithms and logarithmic differnetiation. $$Z=\frac A b\implies \log(Z)=\log(A)-\log(B)$$ making $$\frac{\Delta Z}Z=\frac{\Delta A}A+\frac{\Delta B}B$$ since the errors add. You will for sure notice that this is the same as for $$Z=A\,B$$ .
You can do the same for $$Z=A^a \,B^b\, C^c$$ where the exponents can be positive or negative (or even $$0$$) to get $$\frac{\Delta Z}Z=|a|\frac{\Delta A}A+|b|\frac{\Delta B}B+|c|\frac{\Delta C}C$$