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.