# Power rule

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##### Examples

###### Lessons

- power rule: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$
- constant multiple rule: $\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)$
- $\frac{{d}}{{{d}x}}\left( {{x^{10}} - 5{x^7} + \frac{1}{3}{x^4} - 20{x^3} + {x^2} - 8x - 1000} \right)$

sum rule: $\frac{{d}}{{{d}x}}\left[ {f\left( x \right) + g\left( x \right)} \right] = \frac{{d}}{{{d}x}}f\left( x \right) + \frac{{d}}{{{d}x}}g\left( x \right)$

difference rule: $\frac{{d}}{{{d}x}}\left[ {f\left( x \right) - g\left( x \right)} \right] = \frac{{d}}{{{d}x}}f\left( x \right) - \frac{{d}}{{{d}x}}g\left( x \right)$

- negative exponents: $\frac{1}{x} = {x^{ - 1}}$ and $\frac{1}{{{x^n}}} = {x^{ - n}}$
- rational exponents: $\sqrt x = {x^{\frac{1}{2}}}$ and ${^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}$