Question

If x=ab² , then write relative error in x

Answer 1

Given,

$\sf{\longrightarrow x=ab^2}$

Taking logarithm on both sides,

$\sf{\longrightarrow \log x=\log(ab^2)}$

$\sf{\longrightarrow \log x=\log a+2\log b}$

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

$\sf{\longrightarrow \dfrac{d}{dx}\big[\log x\big]=\dfrac{d}{dx}\big[\log a+2\log b\big]}$

$\sf{\longrightarrow \dfrac{d}{dx}\big[\log x\big]=\dfrac{d}{dx}\big[\log a\big]+2\cdot\dfrac{d}{dx}\big[\log b\big]}$

By chain rule,

$\sf{\longrightarrow \dfrac{d}{dx}\big[\log x\big]=\dfrac{d}{da}\big[\log a\big]\cdot\dfrac{da}{dx}+2\cdot\dfrac{d}{db}\big[\log b\big]\cdot\dfrac{db}{dx}}$

$\sf{\longrightarrow \dfrac{1}{x}=\dfrac{1}{a}\cdot\dfrac{da}{dx}+2\cdot\dfrac{1}{b}\cdot\dfrac{db}{dx}}$

$\sf{\longrightarrow \dfrac{1}{x}=\dfrac{1}{dx}\cdot\dfrac{da}{a}+2\cdot\dfrac{1}{dx}\cdot\dfrac{db}{b}}$

$\sf{\longrightarrow \dfrac{1}{x}=\dfrac{1}{dx}\left[\dfrac{da}{a}+2\cdot\dfrac{db}{b}\right]}$

$\sf{\longrightarrow \dfrac{dx}{x}=\dfrac{da}{a}+2\cdot\dfrac{db}{b}}$

In case of error we can rewrite the equation as,

$\sf{\longrightarrow\underline{\underline{\dfrac{\Delta x}{x}=\dfrac{\Delta a}{a}+2\cdot\dfrac{\Delta b}{b}}}}$

This is the relative error in $\sf{x.}$