Question

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

Answer 1

Given: 

\[f(x) = mx + c\]

Clearly, being a polynomial function, is differentiable everywhere. Therefore the derivative of 

\[f\]at 

\[x\]  is given by:

\[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
\[ \Rightarrow f'(x) = \lim_{h \to 0} \frac{m(x + h) + c - mx - c}{h}\]
\[ \Rightarrow f'(x) = \lim_{h \to 0} \frac{mx + mh + c - mx - c}{h}\]
\[ \Rightarrow f'(x) = \lim_{h \to 0} \frac{mh}{h} \]
\[ \Rightarrow f'(x) = m\]

Thus

\[f'(0) = m\]