# Sum Rule

## Key Questions

• The derivative for $y = f \left(x\right) - g \left(x\right)$ works the same way as the derivative of $y = f \left(x\right) + g \left(x\right)$.

$y = f \left(x\right) - g \left(x\right) \implies \frac{\mathrm{dy}}{\mathrm{dx}} = f ' \left(x\right) - g ' \left(x\right)$

The quick proof is:
$y = f \left(x\right) - g \left(x\right) = f \left(x\right) + \left(- 1\right) g \left(x\right)$
Using the sum rule and the constant rule:
$\frac{\mathrm{dy}}{\mathrm{dx}} = f ' \left(x\right) + \left(- 1\right) g ' \left(x\right) = f ' \left(x\right) - g ' \left(x\right)$.

• By Sum Rule,
$y ' = f ' \left(x\right) + g ' \left(x\right)$

For example, if $y = {x}^{3} + {e}^{x}$, then

$y ' = \left({x}^{3}\right) ' + \left({e}^{x}\right) ' = 3 {x}^{2} + {e}^{x}$