Question

Find the derivative of the constant function $f\left(x\right)=a$ for a fixed real number $a$.

Answer 1

Given, $f\left(x\right)=a$, where $a$ is constant.
We know that $f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{a-a}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{0}{h}$
$\Rightarrow f^{\prime }\left(x\right)=\underset{h\to 0}{lim}0$
$\Rightarrow f^{\prime }\left(x\right)=0$
Hence, derivative of the constant function is $0$.