Proof of Various Derivative Properties — Question 7

Question 7

Prove that the derivative of a constant function is zero. That is, if $$f(x)=c$$ for all $x$, where $c$ is a constant, then $$f'(a)=0$$ for every real number $a$.

Question 7 - Solution

By definition of the derivative, $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.$$

Since $f(x)=c$ for all $x$, we have $$f(a+h)=c \quad\text{and}\quad f(a)=c.$$

Substitute into the difference quotient: $$f'(a) = \lim_{h\to 0}\frac{c-c}{h}.$$

Simplify the numerator: $$f'(a)=\lim_{h\to 0}\frac{0}{h}.$$

Since $\frac{0}{h}=0$ for all $h\neq 0$, the limit is $$f'(a)=0.$$

Therefore, the derivative of a constant function is zero: $$\boxed{f'(a)=0}.$$