vault backup: 2025-02-18 10:01:11

.obsidian/plugins/obsidian-git/data.json

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education/math/MATH1210 (calc 1)/Derivatives.md

@@ -152,15 +152,18 @@ This is used when you want to take the derivative of a function raised to a func
 
 1. $\ln y = \ln (3x \sin x)^{3x}$ 
 2. $\ln y = 3x * ln(2x \sin x)$*
+3. $\dfrac{d}{dx} \ln(y) = \dfrac{d}{dx} 3x(\ln 2 + \ln x + \ln(sinx))$ 
+4. $\dfrac{1}{y} \dfrac{dy}{dx} = 3(\ln 2 + \ln x + \ln(\sin(x))) + 3x (0 + \dfrac{1}{x} + \dfrac{1}{\sin x} * \cos x)$j
+5. $\dfrac{dy}{dx} = (3\ln 2 + 3 \ln x + 3\ln \sin(x) + 3\ln(\sin(x) + 3x\cot(x))(2x\sin x)^{3x}$ 
 # 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 $g(x) = x^2 + 3)$.
-3. First find the derivative of the outside function function ($f(x) = x^4$):
+6. First find the derivative of the outside function function ($f(x) = x^4$):
 $$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$
-4. 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 $x^4$
 
@@ -196,7 +199,7 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$
 - Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$.
 
 Given these facts:
-5. Let $y$ be some function of $x$
-6. $\dfrac{d}{dx} x = 1$
-7. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\
+8. Let $y$ be some function of $x$
+9. $\dfrac{d}{dx} x = 1$
+10. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\