Let a be any constant number.
Then, f(x) = a
f'(x) = \(\lim\limits_{h \to 0}\) \(\frac{f(x+h)-f(x)}{h}\)
We know that coefficient of a linear function is
a = \(\frac{y_2-y_1}{x_2-x_1}\)
Since our function is constant, y1 = y2
Therefore, a = 0
Now,
f'(x) = \(\lim\limits_{h \to 0}\) \(\frac{a-a}{h}\) = \(\lim\limits_{h \to 0}\) \(\frac{0}{h}\) = \(\lim\limits_{h \to 0}\) 0 = 0
Thus, the derivative of a constant function is always 0.