Question

In the equation for a derivative of 1/x the term in the denominator for finding the derivative of as x approaches the limit 0: (x+delta x)x allows us to drop the delta x as it becomes infinitesimally small. What if x became infinitesimally large. While the term (delta x) x would be much smaller than x, it would no longer be insubstantial, or would we just deem it insubstantial in relation to x. Still wouldn't this then be called an inexactitude in our calculation?

Answer 1

you are completely right
we ignore delta x when it is x +delta x bcoz it is very small as compared to x
we can also do so when x is very large then we can ignore even some big values of delta x
what it means geometrically is that in the graph y=1/x the slope remains constan when x is large

For example at x=1000
the slope at x=1000 is -1/x^2 = -1/10^6
this is the instantaneous slope(slope of tangent) delta x is approx = 0
but even a secant line joining x=100 & x=1001 would have the same slope(delta x=1)
slope = y/x=1 = 1/1001 -1/1000 -1/1000*1001
-------------- = -------------- is approx -1/10^6
1001-1000 1