I'm doing approximations using differentiation in one of my classes and was tasked to do the following exercise:
$$ \int_0^x \frac{e^{-t}}{t^{2}+0.51}dt $$
if $x$ changes from $0.7$ to $0.71$.
My textbook has a very bad explanation of similar problems (if you can even consider them explanations) so I am a bit confused. I did the following:
$$ \Delta f=f(0.7)-f(0.71) $$ $$ \Delta f\cong df=\frac{e^{-x}}{x^2+0.51}dx $$
with $x=0.7$ and $dx=0.01$
$$\Rightarrow \Delta f\cong \frac{0.01e^{-0.7}}{0.49+0.51}=\frac{1}{100e^{0.7}}$$
The next step would be to estimate the value of my answer. However, I feel like I've done something wrong and that I should use the same method in such a way to avoid doing two approximations which greatly affect the accuracy of my method.
Can anyone tell me if my approach is correct?
P.S. I would use the Taylor Expansions of $e^x$ to approximate the value of $e^{0.7}$ since a linear approximation at $a=0$ would harm my approximation even more.
For those interested, this is the examples my textbook gives: 
And also this one:
Images were taken from MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES, 3rd ed by MARY L. BOAS.