Answer
Using the definition of the limit we show that $f'(x)=m$. This can be interpreted as the fact that the linear function has a constant slope $m$ for every $x$.
Work Step by Step
We will use the definition of the derivative to calculate $f'(x)$:
$$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{m(x+h)+b-(mx+b)}{h}=\lim_{h\to0}\frac{mx+mh+b-mx-b}{h}=\lim_{h\to0}\frac{mh}{h}=\lim_{h\to0}m=m.$$
The straight line has the property that it is its own tangent at every point. This means that the derivative at every point,which is identified with the slope of the tangent, is constant and equal to that slope. But since the line is its own tangent then this derivative has to be equal to its slope for every $x$ which is $m$ in this case.
