sum rule

The sum rule states that

$$\frac{\mathrm{d}}{\mathrm{d}x}\left[f(x)+g(x)\right]=f^{\prime }(x)+g^{\prime }(x)$$

Proof

See the proof of the sum rule (http://planetmath.org/ProofOfSumRule).

Examples

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(x+1)$$=$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}x+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}1=1$

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(x^2-3x+2)$$=$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}{x}^2+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(-3x)+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(2)=2x-3$

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(\sin x+\cos x)$$=$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}\sin x+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}\cos x=\cos x-\sin x$