Derivative of constant

[suvadip]

If \(\displaystyle f(x)\) is constant then I can show that \(\displaystyle f'(x)=0\).
But if \(\displaystyle f'(x)=0\), then how to show that \(\displaystyle f(x)\) is constant ?

 

[MarkFL]

Re: derivative of constant

What does the definition of the derivative tell you?

edit: Sorry, I misread...have you tried integrating with respect to the independent variable?

 

[Ackbach]

Re: derivative of constant

If \(\displaystyle f(x)\) is constant then I can show that \(\displaystyle f'(x)=0\).
But if \(\displaystyle f'(x)=0\), then how to show that \(\displaystyle f(x)\) is constant ?

Another approach would be to use differentials:
$$\Delta y=f'(x)\, \Delta x = 0 \cdot \Delta x \, ...$$

 

[HallsofIvy]

Re: derivative of constant

If \(\displaystyle f(x)\) is constant then I can show that \(\displaystyle f'(x)=0\).
But if \(\displaystyle f'(x)=0\), then how to show that \(\displaystyle f(x)\) is constant ?

The mean value theorem: if f is continuous and differentiable on [a, b] then there exist c in [a, b] such that [tex]f'(c)= \frac{f(b)- f(a)}{b- a}[/tex]. If f'(x)= 0 for all x, then for any a and b, [itex]\frac{f(b)- f(a)}{b- a}= 0[/tex] from which f(a)= f(b).