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.