Derivative of Constant. Proof and Examples

What is Derivative of constant.

Here we will prove that derivative of the constant is zero.

Let c be constant.

Now by using ab-initio method.

$y=c........(1)$

$y+\delta y=c....(2)$

$eq(2)-eq(1)$

$y+\delta y-y=c-c$

$\delta y=0$

Dividing $\delta x$ on both sides.

$\frac{\delta y}{\delta x}=\frac0{\delta x}$

Applying $\underset{\delta x\rightarrow0}{Lim}$ on both sides.

$\underset{\delta x\rightarrow0}{Lim}\left[\frac{\delta y}{\delta x}\right]=\underset{\delta x\rightarrow0}{Lim}\left[0\right]$

$\boxed{\frac{dy}{dx}=0}$

$f(x)=c$

$f(x+h)=c$

Now by using formula of differentiation.

$f'\left(x\right)=\underset{h\rightarrow0}{Lim}\left[\frac{f(x+h)-f(x)}h\right]$

Now put above values.

$f'\left(x\right)=\underset{h\rightarrow0}{Lim}\left[\frac{c-c}h\right]$

$f'\left(x\right)=\underset{h\rightarrow0}{Lim}\left[\frac0h\right]$

$f'\left(x\right)=\underset{h\rightarrow0}{Lim}\left[0\right]$

$\boxed{f'=0}$

Hence we have to proved that the derivative of constant is zero.