### Differentiation: The derivative of power functions

### The derivative of power functions

In practice, we do not want to calculate derivatives time and time again using the definition of the derivative, but we use calculation rules to directly calculate the derivative. We now first look at the derivative of a power function.

Power rule

\[\dfrac{\dd}{\dd x}(x^\blue{n})=\blue{n}\cdot x^{\blue{n}-1}\]

**Example**

\[\begin{array}{rcl} \dfrac{\dd}{\dd x}(x^\blue{5})&=&\blue{5}\cdot x^{\blue{5}-1}\\&=&\blue{5}x^4\end{array}\]

If we have a power function with a constant in front of it, we can easily factor out the constant.

Power rule with a constant

\[\dfrac{\dd}{\dd x}(\orange{c}\cdot x^\blue{n})=\orange{c}\cdot\blue{n} \cdot x^{\blue{n}-1}\]

**Example**

\[\begin{array}{rcl} \dfrac{\dd}{\dd x}(\orange{2}\cdot x^\blue{-3})&=&\orange{2}\cdot\blue{-3}\cdot x^{\blue{-3}-1}\\&=&-6x^{-4}\end{array}\]

\[\begin{array}{rcl}

\displaystyle \dfrac{\dd y}{\dd x}&=& \displaystyle \dfrac{\dd}{\dd x}\left( {{3}\over{x^5}} \right)\\

&&\phantom{xxx}\blue{y \text{ substituted}}\\

& =&\displaystyle\dfrac{\dd}{\dd x}\left( 3\cdot x^{-5} \right )\\

&&\phantom{xxx}\blue{\text{rewritten to the form }c\cdot x^n}\\

& =& \displaystyle 3 \cdot -5 \cdot x^{-6}\\

&&\phantom{xxx}\blue{\text{power rule with a constant, }\dfrac{\dd}{\dd x}\left (c \cdot x^n\right)=c \cdot n \cdot x^{n-1}}\\

& =&\displaystyle -{{15}\over{x^6}}\\

&&\phantom{xxx}\blue{\text{simplified}}

\end{array}\]

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