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I was just thinking about the question and googled it but couldn't get anything, is it zero because its a constant function or it is anything more complicated??

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    $\begingroup$ It's $0$ because $f(x)=0$ is a constant function. $\endgroup$
    – Rocket Man
    Aug 13, 2015 at 16:54
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    $\begingroup$ In particular, one should learn that differentiation is a linear operator, and applying any linear operator to zero always returns zero. $\endgroup$
    – JMoravitz
    Aug 13, 2015 at 16:58
  • $\begingroup$ Zero has no derivative. The derivative measures the rate of change. Zero does not change, so it can't have a rate of change. $\endgroup$ Aug 13, 2015 at 17:02

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$$f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}h=\lim_{h\to0}\dfrac0h=0.$$

Geometrically speaking, the graph of $f\colon x\mapsto0$ is a horizontal line, so its slope at each point is zero, hence its derivative is equal to zero everywhere. From another perspective, $f\colon x\mapsto0$ is a constant function, it doesn't vary, so its rate of change is zero.

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The point of differentiation is to find the rate of change of a quantity, physically. Since zero never changes, it is clearly a constant. So its derivative must be zero.

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