From 31440675b820a4bc9674f29c06950606eb8746d9 Mon Sep 17 00:00:00 2001 From: arc Date: Thu, 30 Jan 2025 09:28:44 -0700 Subject: [PATCH] vault backup: 2025-01-30 09:28:44 --- education/math/MATH1210 (calc 1)/Derivatives.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/education/math/MATH1210 (calc 1)/Derivatives.md b/education/math/MATH1210 (calc 1)/Derivatives.md index 8c7f900..6818cb6 100644 --- a/education/math/MATH1210 (calc 1)/Derivatives.md +++ b/education/math/MATH1210 (calc 1)/Derivatives.md @@ -56,4 +56,9 @@ $$ f'(x) = \lim_{h \to 0} \dfrac{(x + h)^n - x^n}{h} $$ Note that the coefficient follows the associated level of Pascal's Triangle (`1 3 3 1`), and $x$'s power decrements, while $h$'s power increments. The coefficients of each pair will always add up to $n$. Eg, $3 + 0$, $2 + 1$, $1 + 2$, and so on. The **second** term in the polynomial created will have a coefficient of $n$. -$$ \dfrac{(x + h)^n - x^n}{h} = \lim_{h \to 0} \dfrac{x^n + nx^{n-1}h + p_{n3}x^{n-2}h^2 + \cdots h^n}{} $$ \ No newline at end of file +$$ \dfrac{(x + h)^n - x^n}{h} = \lim_{h \to 0} \dfrac{(x^n + nx^{n-1}h + P_{n3}x^{n-2}h^2 + \cdots + h^n)-x^n}{h} $$ $P$ denotes some coefficient found using Pascal's triangle. + +$x^n$ cancels out, and then $h$ can be factored out of the binomial series. + +This leaves us with: +$$ \lim_{h \to 0} nx^{n-1} + P_{n3} x^{} $$ \ No newline at end of file