Find the derivative of the function f defined by

Given: $f(x)=m x+c$

Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of $f$ at $x$ is given by:

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\Rightarrow f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{m(x+h)+c-m x-c}{h}$

$\Rightarrow f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{m x+m h+c-m x-c}{h}$

$\Rightarrow f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{m h}{h}$

$\Rightarrow f^{\prime}(x)=m$

Thus, $f^{\prime}(0)=m$