Question

Show that if x =a /b then ∆x/x= ∆a/a + ∆b/b

Answer 1

Given,

$\sf{\longrightarrow x=\dfrac{a}{b}}$

Taking logarithm,

$\sf{\longrightarrow \log x=\log\left(\dfrac{a}{b}\right)}$

$\sf{\longrightarrow \log x=\log a-\log b}$

Differentiating with respect to $\sf{x,}$

$\sf{\longrightarrow \dfrac{d}{dx}(\log x)=\dfrac{d}{dx}(\log a-\log b)}$

$\sf{\longrightarrow \dfrac{1}{x}=\dfrac{1}{a}\cdot\dfrac{da}{dx}-\dfrac{1}{b}\cdot\dfrac{db}{dx}}$

$\sf{\longrightarrow \dfrac{1}{x}=\dfrac{1}{dx}\cdot\dfrac{da}{a}-\dfrac{1}{dx}\cdot\dfrac{db}{b}}$

$\sf{\longrightarrow \dfrac{1}{x}=\dfrac{1}{dx}\left[\dfrac{da}{a}-\dfrac{db}{b}\right]}$

$\sf{\longrightarrow \dfrac{dx}{x}=\dfrac{da}{a}-\dfrac{db}{b}}$

In case of interval change,

$\sf{\longrightarrow \dfrac{\Delta x}{x}=\dfrac{\Delta a}{a}-\dfrac{\Delta b}{b}}$

In case of maximum possible error, the negative sign should be replaced by positive sign else in order to obtain maximum possible error, irrespective of the sign of power of $\sf{b.}$

That's why,

$\sf{\longrightarrow\underline{\underline{\dfrac{\Delta x}{x}=\dfrac{\Delta a}{a}+\dfrac{\Delta b}{b}}}}$

Thus proved!