Question

Find the derivative of the function f defined by f (x) = mx + c at x = 0.

Answer 1

Given: $f\left(x\right)=mx+c$

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

$$\begin{gathered} f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ \Rightarrow f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{m(x+h)+c-mx-c}{h} \\ \Rightarrow f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{mx+mh+c-mx-c}{h} \\ \Rightarrow f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{mh}{h} \\ \Rightarrow f\text{'}\left(x\right)=m \end{gathered}$$

Thus, $f\text{'}\left(0\right)=m$