How do I use the limit definition of derivative to find #f'(x)# for #f(x)=mx+b# ?

1 Answer
Aug 29, 2014

Remember that the limit definition of the derivative goes like this:
#f'(x)=lim_{h rightarrow0}{f(x+h)-f(x)}/{h}#.
So, for the posted function, we have
#f'(x)=lim_{hrightarrow0}{m(x+h)+b-[mx+b]}/{h}#
By multiplying out the numerator,
#=lim_{hrightarrow0}{mx+mh+b-mx-b}/{h}#
By cancelling out #mx#'s and #b#'s,
#=lim_{hrightarrow0}{mh}/{h}#
By cancellng out #h#'s,
#=lim_{hrightarrow0}m=m#
Hence, #f'(x)=m#.

The answer above makes sense since the derivative tells us about the slope of the tangent line to the graph of #f#, and the slope of the linear function (its graph is a line) is #m#.