In Section CCD.DDN4S there is a mapping diagram using the four step method used to find the derivatives for $f(x)=1$, $f(x)=x, f(x)=x^2,$ and $f(x)=\frac1x = x^{-1} $.

These results are summarized in the following table:$f(x)$ | $f'(x)$ |

$1$ | $0$ |

$x$ | $1$ |

$x^2$ | $2x$ |

$x^3$ | $3x^2$ |

$\frac 1x=x^{-1}$ | $\frac {-1}{x^2}=(-1)x^{-2}, x\ne 0$ |

It is time to think of these derivatives for power functions a little more generally.

In Theorem CCD.DP we state the most general result, but in this section we cover only powers that are integers. Discussion of rational and real number powers are found in later sections using the inverse relation for roots and the chain rule for real number powers connected to the natural exponential and logarithm functions.

Details for the proofs can be found in most calculus texts or in

What is missing in other proofs are visualizations using mapping diagrams. These are provided with the mapping diagram CCD.DPN.0 and CCD.DPMD.

Theorem CCD.DP

then the derivative of $f(x)=x^p$ is $$f'(x)=D_x(x^p)=\frac {d\ (x^p)}{dx} =px^{p-1}$$ for any $x$ -

with the

**Theorem CCD.DPN** If $n$ is any natural number,

then the derivative of
$f(x)=x^n$ is $$f'(x)=D_x(x^n)=\frac {d\ (x^n)}{dx} =nx^{n-1}$$ for any
$x$;

Check this visually with the mapping diagram **CDD.DPN.0**

CDD.DPN.0

Download GeoGebra fileCDD.DPN.0

Now that we have found the derivative for $f(x) = x^n$ for $ n =0,1,2,3,...$, it is time to think of these derivatives for negative integer values for $n$.

For detailed proofs of this result see almost any calculus text or

CDD.DPZ |