Define infinitesimally small? Why do we ignore squares of infinitesimal quantities in differential calculus , doesn’t that cause error when we require very accurate measurements.?
when you find a derivative it is something like $\frac{dy}{dx}=\frac{a_0\ +\ a_1{dx}\ +\  a_2{dx}^2 .......}{dx}$. Here $a_0\ +\ a_1\ .....$can be a function of x. So if $a_0$ is not zero this thing will tend to infinity (so lets us assume it to be zero). Overall it is $\frac{dy}{dx}=a_1\ +\ a_2{dx}.... $ . Now obviously you can take dx to be very small which is tending to zero , which will not give error.(let us say u r calculating till one billionth decimal place , you can take dx to be even smaller than that)
In all empirical sciences there is error no matter how accurate the measurements are. They will always be rational numbers and the error term is always non-trivial. You have to justify ignoring infinitesimals and that typically involves showing the error term is smaller than the error of the measurement. in practice this happens routinely and so it's useful to study the case of negligible terms.