From 763c9022ca375271ebffd8e26fe041cd2c639686 Mon Sep 17 00:00:00 2001
From: arc <zleyyij@users.noreply.github.com>
Date: Sun, 16 Feb 2025 18:52:21 -0700
Subject: [PATCH] vault backup: 2025-02-16 18:52:21

---
 .obsidian/plugins/obsidian-git/data.json      | 27 +++++++++++++++++++
 .../math/MATH1210 (calc 1)/Derivatives.md     | 20 +++++++++-----
 2 files changed, 41 insertions(+), 6 deletions(-)

diff --git a/.obsidian/plugins/obsidian-git/data.json b/.obsidian/plugins/obsidian-git/data.json
index e69de29..bef4c6e 100644
--- a/.obsidian/plugins/obsidian-git/data.json
+++ b/.obsidian/plugins/obsidian-git/data.json
@@ -0,0 +1,27 @@
+{
+  "commitMessage": "vault backup: {{date}}",
+  "autoCommitMessage": "vault backup: {{date}}",
+  "commitDateFormat": "YYYY-MM-DD HH:mm:ss",
+  "autoSaveInterval": 5,
+  "autoPushInterval": 0,
+  "autoPullInterval": 5,
+  "autoPullOnBoot": true,
+  "disablePush": false,
+  "pullBeforePush": true,
+  "disablePopups": false,
+  "listChangedFilesInMessageBody": false,
+  "showStatusBar": true,
+  "updateSubmodules": false,
+  "syncMethod": "merge",
+  "customMessageOnAutoBackup": false,
+  "autoBackupAfterFileChange": false,
+  "treeStructure": false,
+  "refreshSourceControl": true,
+  "basePath": "",
+  "differentIntervalCommitAndPush": false,
+  "changedFilesInStatusBar": false,
+  "showedMobileNotice": true,
+  "refreshSourceControlTimer": 7000,
+  "showBranchStatusBar": true,
+  "setLastSaveToLastCommit": false
+}
\ No newline at end of file
diff --git a/education/math/MATH1210 (calc 1)/Derivatives.md b/education/math/MATH1210 (calc 1)/Derivatives.md
index cd116f7..90fa3d5 100644
--- a/education/math/MATH1210 (calc 1)/Derivatives.md	
+++ b/education/math/MATH1210 (calc 1)/Derivatives.md	
@@ -121,7 +121,7 @@ $$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$
 > Apply the chain rule to $x^4$
 
 If we treat the above as a function along the lines of $f(x) = (x)^4$, and $g(x) = x$, then the chain rule can be used like so:
-$$ 4(x)^3 * x $$
+$$ 4(x)^3 * (1) $$
 # Trig Functions
 $$ \lim_{x \to 0} \dfrac{\sin x}{x} = 1 $$
 $$ \lim_{x \to 0} \dfrac{\cos x - 1}{x} = 0 $$
@@ -150,12 +150,20 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$
 - $\dfrac{d}{dx} x = \dfrac{dx}{dx} = 1$ : The derivative of $x$ with respect to $x$ is one
 - $\dfrac{d}{dx} y = \dfrac{dy}{dx} = y'$ 
 - Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$.
+
+Given these facts:
+1. Let $y$ be some function of $x$
+2. $\dfrac{d}{dx} x = 1$
+3. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\
+What's the derivative of $y^2$?
+$\dfrac{d}{dx} y^2 = 2(y)^1 *\dfrac{dy}{dx}$ 
+
 # Examples
 
 > Differentiate $f(x) = 4\sqrt[3]{x} - \dfrac{1}{x^6}$ 
 
-2. $f(x) = 4\sqrt[3]{x} = \dfrac{1}{x^6}$
-3. $= 4x^\frac{1}{3} - x^{-6}$  
-4. $f'(x) = \dfrac{1}{3} * 4x^{-\frac{2}{3}} -(-6)(x^{-6-1})$ 
-5. $= 4x^{-2-\frac{2}{3}} + 6x^{-7}$
-6. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$
\ No newline at end of file
+4. $f(x) = 4\sqrt[3]{x} = \dfrac{1}{x^6}$
+5. $= 4x^\frac{1}{3} - x^{-6}$  
+6. $f'(x) = \dfrac{1}{3} * 4x^{-\frac{2}{3}} -(-6)(x^{-6-1})$ 
+7. $= 4x^{-2-\frac{2}{3}} + 6x^{-7}$
+8. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$
\ No newline at end of file