Why is derivative of a sum the sum of the derivatives?

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Using the definition of derivative, we show that the derivative of a sum is the sum of the derivatives, provided the derivatives exist.
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Love this! Exactly what I was looking for, thanks!

iliasaarab
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One of these days I hope to be able to describe calculus concepts this clearly. This is my first year teaching it so I have a long way to go.

joshuawalker
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exactly, way too many ppl teach this without specifying that the theorem is for a specific point(because its a limit), not a variable. I still have no idea how to get to f'(x) as a variable except using intuition with the domain of the graph as a guide, it baffles me how no one understands this crucial difference and blindly applies this 'formula'

xuepingsong