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

Aug 29, 2014

Remember that the limit definition of the derivative goes like this:
$f ' \left(x\right) = {\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$.
So, for the posted function, we have
$f ' \left(x\right) = {\lim}_{h \rightarrow 0} \frac{m \left(x + h\right) + b - \left[m x + b\right]}{h}$
By multiplying out the numerator,
$= {\lim}_{h \rightarrow 0} \frac{m x + m h + b - m x - b}{h}$
By cancelling out $m x$'s and $b$'s,
$= {\lim}_{h \rightarrow 0} \frac{m h}{h}$
By cancellng out $h$'s,
$= {\lim}_{h \rightarrow 0} m = m$
Hence, $f ' \left(x\right) = m$.

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 $m$.