Studying the use of difference quotient when solving for functions of power.
The difference quotient is
\[f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\]and solving for the derivative of say \( \frac{1}{x} \) or \( \sin(x) \) is straightforward by substitution and algebraic manipulation.
Things change for solving the derivative of power functions and the difference quotient form becomes
\[f'(x)=\lim_{\Delta x \to 0}\frac{f((x+\Delta x)^n)-f(x^n)}{\Delta x}\]and the binomial theorem becomes handy
\[(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\]Using the binomial theorem in the context of the difference equation, the form can be expanded as
\[(x+\Delta x)^n=x^n+nx^{n-1}\Delta x+\frac{n(n-1)}{2}x^{n-2}(\Delta x)^2+\frac{n(n-1)(n-2)}{6}x^{n-3}(\Delta x)^3+\cdots\]and I just learned that the third term onwards disappears as \( \Delta x \) approaches zero. These terms are represented as \( O((\Delta x)^2) \). The binomial expansion form becomes
\[(x+\Delta x)^n=x^n+nx^{n-1}\Delta x+O((\Delta x)^2)\]This site will hold short thoughts, daily observations, technical discoveries, project progress, and records that do not require a full article.