PlanetMath: sum rule

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sum rule

(Theorem)

The sum rule states that

$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}\left[f(x)+g(x)\right] = f'(x) + g'(x)$

Examples

$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}(x + 1)$	$\displaystyle =$	$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}x + \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}1 = 1$
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}(x^2 - 3x + 2)$	$\displaystyle =$	$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}... ...\ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}(2) = 2x-3$
$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}(\sin x + \cos x)$	$\displaystyle =$	$\displaystyle \ensuremath{\frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}... ...rac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}x}}}\cos x = \cos x - \sin x$

"sum rule" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Attachments:: proof of sum rule (Proof) by mathcam

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This is version 5 of sum rule, born on 2002-02-24, modified 2004-03-22.
Object id is 2637, canonical name is SumRule.
Accessed 4821 times total.

Classification:

AMS MSC:

26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)

Pending Errata and Addenda

None.[ View all 2 ]

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Proof

Examples