# Derivative of Constant. Proof and Examples

## What is Derivative of constant.

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

## Method 1

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}

## Method 2

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. 