diff --git a/.obsidian/plugins/obsidian-git/data.json b/.obsidian/plugins/obsidian-git/data.json index bef4c6e..e69de29 100644 --- a/.obsidian/plugins/obsidian-git/data.json +++ b/.obsidian/plugins/obsidian-git/data.json @@ -1,27 +0,0 @@ -{ - "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 94f0627..cd116f7 100644 --- a/education/math/MATH1210 (calc 1)/Derivatives.md +++ b/education/math/MATH1210 (calc 1)/Derivatives.md @@ -115,8 +115,13 @@ $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$ Using the chain rule, the above function might be described as $f(g(x))$, where $f(x) = x^4$, and $g(x) = x^2 + 3)$. 1. First find the derivative of the outside function function ($f(x) = x^4$): -$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 $$ -2. +$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$ +2. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. +$$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$ +> 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: +$$ 4(x)^3 * x $$ # Trig Functions $$ \lim_{x \to 0} \dfrac{\sin x}{x} = 1 $$ $$ \lim_{x \to 0} \dfrac{\cos x - 1}{x} = 0 $$ @@ -149,8 +154,8 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$ > Differentiate $f(x) = 4\sqrt[3]{x} - \dfrac{1}{x^6}$ -3. $f(x) = 4\sqrt[3]{x} = \dfrac{1}{x^6}$ -4. $= 4x^\frac{1}{3} - x^{-6}$ -5. $f'(x) = \dfrac{1}{3} * 4x^{-\frac{2}{3}} -(-6)(x^{-6-1})$ -6. $= 4x^{-2-\frac{2}{3}} + 6x^{-7}$ -7. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$ \ No newline at end of file +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