From f4c9265e07565c14c962f7b24333d101e0d77029 Mon Sep 17 00:00:00 2001 From: arc Date: Tue, 18 Feb 2025 10:11:11 -0700 Subject: [PATCH] vault backup: 2025-02-18 10:11:11 --- .../math/MATH1210 (calc 1)/Derivatives.md | 23 ++++++++++--------- 1 file changed, 12 insertions(+), 11 deletions(-) diff --git a/education/math/MATH1210 (calc 1)/Derivatives.md b/education/math/MATH1210 (calc 1)/Derivatives.md index f4c1b3a..f16111c 100644 --- a/education/math/MATH1210 (calc 1)/Derivatives.md +++ b/education/math/MATH1210 (calc 1)/Derivatives.md @@ -152,24 +152,25 @@ This is used when you want to take the derivative of a function raised to a func 1. $\ln y = \ln((7x+2)^x)$ 2. $\ln y = x*\ln(7x + 2)$ -3. $\dfrac{dy}{dx} \dfrac{1}{y} = 3x*\dfrac{1}{7x + 2} * 3\ln(7x+2)$ +3. $\dfrac{dy}{dx} \dfrac{1}{y} = \dfrac{7x}{7x + 2} * \ln(7x+2)$ +4. $\dfrac{dy}{dx} = (\dfrac{7x}{7x+2} * \ln(7x+2))\ln(7x+2)^x$ > Find the derivative of the function $y = (2x \sin x)^{3x}$ -4. $\ln y = \ln (3x \sin x)^{3x}$ -5. $\ln y = 3x * \ln(2x \sin x)$* -6. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$ -7. $\dfrac{1}{y} \dfrac{dy}{dx} = 3(\ln 2 + \ln x + \ln(\sin(x))) + 3x (0 + \dfrac{1}{x} + \dfrac{1}{\sin x} * \cos x)$j -8. $\dfrac{dy}{dx} = (3\ln 2 + 3 \ln x + 3\ln \sin(x) + 3\ln(\sin(x) + 3x\cot(x))(2x\sin x)^{3x}$ +5. $\ln y = \ln (3x \sin x)^{3x}$ +6. $\ln y = 3x * \ln(2x \sin x)$* +7. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$ +8. $\dfrac{1}{y} \dfrac{dy}{dx} = 3(\ln 2 + \ln x + \ln(\sin(x))) + 3x (0 + \dfrac{1}{x} + \dfrac{1}{\sin x} * \cos x)$j +9. $\dfrac{dy}{dx} = (3\ln 2 + 3 \ln x + 3\ln \sin(x) + 3\ln(\sin(x) + 3x\cot(x))(2x\sin x)^{3x}$ # 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 $g(x) = x^2 + 3)$. -9. First find the derivative of the outside function function ($f(x) = x^4$): +10. First find the derivative of the outside function function ($f(x) = x^4$): $$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$ -10. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. +11. 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 $x^4$ @@ -205,7 +206,7 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$ - Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$. Given these facts: -11. Let $y$ be some function of $x$ -12. $\dfrac{d}{dx} x = 1$ -13. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ +12. Let $y$ be some function of $x$ +13. $\dfrac{d}{dx} x = 1$ +14. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\