$\large\frac{1}{10.1}=\frac{1}{10+0.1}$

Step 1:

Let $y=f(x)=\large\frac{1}{x}$

$dy=-\large\frac{1}{x^2}$$dx$

Step 2:

Let $x=10,dx \approx \Delta x=0.1$

Then $dy=-\large\frac{-1}{100}$$ \times 0.1=-0.001$

We also have $f(10)=\large\frac{1}{10}$$+dy$

Now $f(10.1)=f(10)+dy$

$\qquad\qquad=0.1-0.001$

$\qquad\qquad=0.099$