# Power rule

### Power rule

When using the Definition of Derivative, finding the derivative of a long polynomial function with large exponents, or powers, can be very demanding. To avoid this, we introduce you one of the most powerful differentiation tools that simplifies this entire differentiation process – the Power Rule. In this section, we will see how the Power Rule allows us to easily derive the slope of a polynomial function at any given point.

#### Lessons

POWER RULE: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$ , where $n$ is any real number
• 1.
power rule: $\frac{{d}}{{{d}x}}\left( {{x^n}} \right) = n\;{x^{n - 1}}$
a)
${\;}\frac{{d}}{{{d}x}}\left( {{x^5}} \right)$

b)
${\;}\frac{{d}}{{{d}x}}\left( x \right)$

c)
${\;}\frac{{d}}{{{d}x}}\left( 3 \right)$

• 2.
constant multiple rule: $\frac{{d}}{{{d}x}}\left[ {cf\left( x \right)} \right] = c\;\frac{{d}}{{{d}x}}f\left( x \right)$
a)
${\;}\frac{{d}}{{{d}x}}\left( {4{x^3}} \right)$

b)
${\;}\frac{{d}}{{{d}x}}\left( {6x} \right)$

c)
${\;}\frac{{d}}{{{d}x}}\left( { - x} \right)$

• 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 $\frac{1}{{{x^n}}} = {x^{ - n}}$
a)
${\;}\frac{{d}}{{{d}x}}\left( {\frac{1}{{{x^2}}}} \right)$

b)
${\;}\frac{{d}}{{{d}x}}\left( {\frac{{ - 5}}{{3x}}} \right)$

• 5.
rational exponents: $\sqrt x = {x^{\frac{1}{2}}}$ and ${^b}\sqrt{{{x^a}}} = {x^{\frac{a}{b}}}$
a)
${\;}\frac{{d}}{{{d}x}}\left( {{^3}\sqrt{{{x^5}}}} \right)$

b)
${\;}\frac{{d}}{{{d}x}}\left( {\sqrt x } \right)$

c)
${\;}\frac{{d}}{{{d}x}}\left( {\frac{8}{{21\sqrt {{x^3}} }}} \right)$