Question

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

Answer 1

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`.