Derivation of Errors on Physical Quantities

If two physical quantities x and y are related as y=kxⁿ if the percentage error on x is P derive an expression to find the percentage error on y

y=kxⁿ------------------------(1)

Δy=k n x^(n-1) Δx -----------(2) (by differentiating  equation 1)

Divide equation (2) by (1)

Δy /y =n Δx/ x

(Δy /y) ×100=n  (Δx/ x) ×100

ie percentage error on y =n times percentage error on x = n P

Note that absolute eror in 'y' is denoted by Δy , relative error by Δy /y and percentage error by  (Δy /y)×100

so the general rules are if
i) y=a+b or a-b
then absolute error in 'y' is the sum of the absolute errors of 'a' and 'b'
ie Δy=Δa+Δb

ii) y=a× b or a/b 
then the relative error in 'y ' is the sum of the relative errors of 'a' and 'b'
ie Δy /y=Δa /a+Δb /b
so same is the percentage error in 'y'

iii) y=kxⁿ
then the relative error in 'y ' is n times relative  error on x 
so the percentage error also

Prepared by Sai’s Classes Poonkunnam Thrissur +919846088761