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.
[katex]y=c……..(1)[/katex]
[katex]y+\delta y=c….(2)[/katex]
[katex]eq(2)-eq(1)[/katex]
[katex]y+\delta y-y=c-c[/katex]
[katex]\delta y=0[/katex]
Dividing [katex]\delta x[/katex] on both sides.
[katex]\frac{\delta y}{\delta x}=\frac0{\delta x}[/katex]
Applying [katex]\underset{\delta x\rightarrow0}{Lim}[/katex] on both sides.
[katex]\underset{\delta x\rightarrow0}{Lim}\left[\frac{\delta y}{\delta x}\right]=\underset{\delta x\rightarrow0}{Lim}\left[0\right][/katex]
[katex]\boxed{\frac{dy}{dx}=0}[/katex]
Method 2
[katex]f(x)=c[/katex]
[katex]f(x+h)=c[/katex]
Now by using formula of differentiation.
[katex]f’\left(x\right)=\underset{h\rightarrow0}{Lim}\left[\frac{f(x+h)-f(x)}h\right][/katex]
Now put above values.
[katex]f’\left(x\right)=\underset{h\rightarrow0}{Lim}\left[\frac{c-c}h\right][/katex]
[katex]f’\left(x\right)=\underset{h\rightarrow0}{Lim}\left[\frac0h\right][/katex]
[katex]f’\left(x\right)=\underset{h\rightarrow0}{Lim}\left[0\right][/katex]
[katex]\boxed{f’=0}[/katex]
Hence we have to proved that the derivative of constant is zero.