derivative of the regular polygon formula

the derivative of the regular polygon formula found in the process of finding the equable regular polygon formula, A_P, may be found by taking the partial derivative with respect to s:

$\displaystyle{\frac{\partial}{\partial s}}$ $A_P(s)$

which is evaluated by the following:

$\displaystyle\frac{\partial}{\partial s}}$ $A_P(s)$ $=\frac{n\cot\frac{\pi}{n}}{4}(\displaystyle\frac{\partial}{\partial s}s^2)$

$\displaystyle\frac{\partial}{\partial s}s^2=2s$

$\displaystyle\frac{\partial}{\partial s}}$ $A_P(s)=$ $\frac{2ns\cot\frac{\pi}{n}}{4}$

$\displaystyle\frac{\partial}{\partial s}}$ $A_P(s)=$ $\frac{ns\cot\frac{\pi}{n}}{2}$

this derivative is interesting as the rate of change of the area of a polygon and its area are numerically equal when s = 2:

$\displaystyle\frac{\partial}{\partial s}}$ $A_P(2)=$ $\frac{2n\cot\frac{\pi}{n}}{2}\quad,\quad$ ${A_P(2)=$ $\frac{2^2n\cot\frac{\pi}{n}}{4}$

$\displaystyle\frac{\partial}{\partial s}}$ $A_P(2)=$ $n\cot\frac{\pi}{n}\quad,\quad$ ${A_P(2)=$ $n\cot\frac{\pi}{n}$

which means that, for any regular polygon with side length s and side count n, the area of the polygon is equal to the rate of change of the area of the polygon when the side length of the polygon is equal to two
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