Question

Show that constant function is always differentiable

Answer 1

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.