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vault backup: 2025-02-18 09:56:11

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arc 2025-02-18 09:56:11 -07:00
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27
.obsidian/plugins/obsidian-git/data.json vendored
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@ -1,27 +0,0 @@
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25
education/math/MATH1210 (calc 1)/Derivatives.md
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@ -139,19 +139,28 @@ This is used when you want to take the derivative of a function raised to a func
1. $\dfrac{d}{dx} x^x$ 1. $\dfrac{d}{dx} x^x$
2. $y = x^x$ 2. $y = x^x$
3. $\ln y = \ln x^x$ 3. Take the natural log of both sides: $\ln y = \ln x^x$
4. $\ln(y) = x*\ln(x)$ 4. $\ln(y) = x*\ln(x)$
5. $\dfrac{d}{dx} \ln y = \dfrac{d}{dx} x \ln x$ 5. Use implicit differentiation: $\dfrac{d}{dx} \ln y = \dfrac{d}{dx} x \ln x$
6. $\dfrac{1}{y} \dfrac{dy}{dx} = 1 * \ln x + x * \dfrac{1}{x}$ 6. Solve for $\dfrac{dy}{dx}$: $\dfrac{1}{y} \dfrac{dy}{dx} = 1 * \ln x + x * \dfrac{1}{x}$
7. $\dfrac{dy}{dx} = (\ln x + 1) * y$
8. Referring back to step 2, $y = x^x$, so the final form is:
9. $\dfrac{dy}{dx} = (\ln(x) + 1)x^x$
### Examples
> Find the derivative of the function $y = (2x \sin x)^{3x}$
1. $\ln y = \ln (3x \sin x)^{3x}$
2. $\ln y = 3x * ln(2x \sin x)$*
# Chain Rule # Chain Rule
$$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$ $$ \dfrac{d}{dx} f(g(x)) = f'(g(x))*g'(x) $$
## Examples ## Examples
> Given the function $(x^2+3)^4$, find the derivative. > 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 $g(x) = x^2 + 3)$. Using the chain rule, the above function might be described as $f(g(x))$, where $f(x) = x^4$, and $g(x) = x^2 + 3)$.
7. First find the derivative of the outside function function ($f(x) = x^4$): 3. First find the derivative of the outside function function ($f(x) = x^4$):
$$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$ $$ \dfrac{d}{dx} (x^2 +3)^4 = 4(g(x))^3 ...$$
8. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$. 4. Multiply that by the derivative of the inside function, $g(x)$, or $x^2 + 3$.
$$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$ $$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$
> Apply the chain rule to $x^4$ > Apply the chain rule to $x^4$
@ -187,7 +196,7 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$
- Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$. - Given the equation $y = x^2$, $\dfrac{d}{dx} y = \dfrac{dy}{dx} = 2x$.
Given these facts: Given these facts:
9. Let $y$ be some function of $x$ 5. Let $y$ be some function of $x$
10. $\dfrac{d}{dx} x = 1$ 6. $\dfrac{d}{dx} x = 1$
11. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\ 7. $\dfrac{d}{dx} y = \dfrac{dy}{dx}$\