CCD.DP: The derivatives of the power functions: integer powers. [See Sensible Calculus]

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 The Sensible Calculus Book Section I.F Finding the Derivative of Some Key Functions.
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
If $p$ is any real number,
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 exception that $f$ is not differentiable at $x=0$  when $p-1<0$.

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 file



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 The Sensible Calculus Book Section I.F Finding the Derivative of Some Key Functions.



CDD.DPMD
Download GeoGebra file