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 90fa3d5..03f5542 100644 --- a/education/math/MATH1210 (calc 1)/Derivatives.md +++ b/education/math/MATH1210 (calc 1)/Derivatives.md @@ -15,6 +15,15 @@ If we have the coordinate pair $(a, f(a))$, and the value $h$ is the distance be $$\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h}$$ The above formula can be used to find the *derivative*. This may also be referred to as the *instantaneous velocity*, or the *instantaneous rate of change*. +## Examples + +> Differentiate $f(x) = 4\sqrt[3]{x} - \dfrac{1}{x^6}$ + +1. $f(x) = 4\sqrt[3]{x} = \dfrac{1}{x^6}$ +2. $= 4x^\frac{1}{3} - x^{-6}$ +3. $f'(x) = \dfrac{1}{3} * 4x^{-\frac{2}{3}} -(-6)(x^{-6-1})$ +4. $= 4x^{-2-\frac{2}{3}} + 6x^{-7}$ +5. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$ # Point Slope Formula (Review) $$ y - y_1 = m(x-x_1) $$ Given that $m = f'(a)$ and that $(x_1, y_1) = (a, f(a))$, you get the equation: @@ -114,9 +123,9 @@ $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$ > Given the function $(x^2+3)^4$, find the derivative. 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 ($f(x) = x^4$): +6. 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. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. +7. 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 $x^4$ @@ -152,18 +161,8 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$ - 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 $y^2$? -$\dfrac{d}{dx} y^2 = 2(y)^1 *\dfrac{dy}{dx}$ +8. Let $y$ be some function of $x$ +9. $\dfrac{d}{dx} x = 1$ +10. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ -# Examples - -> Differentiate $f(x) = 4\sqrt[3]{x} - \dfrac{1}{x^6}$ - -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 +## Examples