From acf45de1603a1507c8330e337200f2b27598ffd1 Mon Sep 17 00:00:00 2001 From: arc Date: Sun, 16 Feb 2025 18:42:21 -0700 Subject: [PATCH] vault backup: 2025-02-16 18:42:21 --- education/math/MATH1210 (calc 1)/Derivatives.md | 16 +++++++++++----- 1 file changed, 11 insertions(+), 5 deletions(-) diff --git a/education/math/MATH1210 (calc 1)/Derivatives.md b/education/math/MATH1210 (calc 1)/Derivatives.md index e22db1e..94f0627 100644 --- a/education/math/MATH1210 (calc 1)/Derivatives.md +++ b/education/math/MATH1210 (calc 1)/Derivatives.md @@ -110,7 +110,13 @@ for all $a > 0$ # Chain Rule $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$ +## Examples +> 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 ($f(x) = x^4$): +$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 $$ +2. # Trig Functions $$ \lim_{x \to 0} \dfrac{\sin x}{x} = 1 $$ $$ \lim_{x \to 0} \dfrac{\cos x - 1}{x} = 0 $$ @@ -143,8 +149,8 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$ > 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. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$ \ No newline at end of file +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