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.

Derivative of constant

[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.

learn product rule of derivative

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