Power rule

Lessons

1. power rule: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$
   1. ${\;}\frac{{d}}{{{d}x}}\left( {{x^5}} \right)$
   2. ${\;}\frac{{d}}{{{d}x}}\left( x \right)$
   3. ${\;}\frac{{d}}{{{d}x}}\left( 3 \right)$
2. constant multiple rule: $\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)$
   1. ${\;}\frac{{d}}{{{d}x}}\left( {4{x^3}} \right)$
   2. ${\;}\frac{{d}}{{{d}x}}\left( {6x} \right)$
   3. ${\;}\frac{{d}}{{{d}x}}\left( { - x} \right)$
3. $\frac{{d}}{{{d}x}}\left( {{x^{10}} - 5{x^7} + \frac{1}{3}{x^4} - 20{x^3} + {x^2} - 8x - 1000} \right)$
   
   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: $\frac{{d}}{{{d}x}}\left[ {f\left( x \right) - g\left( x \right)} \right] = \frac{{d}}{{{d}x}}f\left( x \right) - \frac{{d}}{{{d}x}}g\left( x \right)$
   
4. negative exponents: $\frac{1}{x} = {x^{ - 1}}$ and $\frac{1}{{{x^n}}} = {x^{ - n}}$
   1. ${\;}\frac{{d}}{{{d}x}}\left( {\frac{1}{{{x^2}}}} \right)$
   2. ${\;}\frac{{d}}{{{d}x}}\left( {\frac{{ - 5}}{{3x}}} \right)$
5. rational exponents: $\sqrt x = {x^{\frac{1}{2}}}$ and ${^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}$
   1. ${\;}\frac{{d}}{{{d}x}}\left( {{^3}\sqrt{{{x^5}}}} \right)$
   2. ${\;}\frac{{d}}{{{d}x}}\left( {\sqrt x } \right)$
   3. ${\;}\frac{{d}}{{{d}x}}\left( {\frac{8}{{21\sqrt {{x^3}} }}} \right)$