Question

If f= x3, then relative error in ‘f’ would be how many times the relative error in x?

Answer 1

Given,

$\longrightarrow f=x^3\quad\quad\dots(1)$

The relative error in $f$ is $\dfrac{df}{f}$ and that in $x$ is $\dfrac{dx}{x}.$

Differentiating (1) with respect to $x,$

$\longrightarrow\dfrac{df}{dx}=3x^2$

$\longrightarrow df=3x^2\ dx\quad\quad\dots(2)$

Dividing (2) by (1),

$\longrightarrow\dfrac{df}{f}=\dfrac{3x^2\ dx}{x^3}$

$\longrightarrow\underline{\underline{\dfrac{df}{f}=3\cdot \dfrac{dx}{x}}}$

I.e., the relative error in $f$ is 3 times that in $x.$