Question:
If $y=\log _{e} x$, then find $\Delta y$ when $x=3$ and $\Delta x=0.03$.
Solution:
We have
$x=3$
$\Delta x=0.03$
$y=\log _{e} x$
For $x=3$,
$y=\log _{e} 3$
Also, $\frac{d y}{d x}=\frac{1}{x}$
$\Rightarrow\left(\frac{d y}{d x}\right)_{x=3}=\frac{1}{3}$
$\Rightarrow \Delta y=d y=\frac{d y}{d x} d x=\frac{1}{3} \times 0.03=0.01$