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vault backup: 2025-02-16 18:52:21

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arc 2025-02-16 18:52:21 -07:00
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27
.obsidian/plugins/obsidian-git/data.json vendored
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@ -0,0 +1,27 @@
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20
education/math/MATH1210 (calc 1)/Derivatives.md
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@ -121,7 +121,7 @@ $$ \dfrac{d}{dx} (x^2 + 3)^4 = 4(x^2 + 3)^3 * (2x)$$
> Apply the chain rule to $x^4$ > Apply the chain rule to $x^4$
If we treat the above as a function along the lines of $f(x) = (x)^4$, and $g(x) = x$, then the chain rule can be used like so: If we treat the above as a function along the lines of $f(x) = (x)^4$, and $g(x) = x$, then the chain rule can be used like so:
$$ 4(x)^3 * x $$ $$ 4(x)^3 * (1) $$
# Trig Functions # Trig Functions
$$ \lim_{x \to 0} \dfrac{\sin x}{x} = 1 $$ $$ \lim_{x \to 0} \dfrac{\sin x}{x} = 1 $$
$$ \lim_{x \to 0} \dfrac{\cos x - 1}{x} = 0 $$ $$ \lim_{x \to 0} \dfrac{\cos x - 1}{x} = 0 $$
@ -150,12 +150,20 @@ $$ \dfrac{d}{dx} \cot x = -\csc^2 x $$
- $\dfrac{d}{dx} x = \dfrac{dx}{dx} = 1$ : The derivative of $x$ with respect to $x$ is one - $\dfrac{d}{dx} x = \dfrac{dx}{dx} = 1$ : The derivative of $x$ with respect to $x$ is one
- $\dfrac{d}{dx} y = \dfrac{dy}{dx} = y'$ - $\dfrac{d}{dx} y = \dfrac{dy}{dx} = y'$
- 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:
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 $y^2$?
$\dfrac{d}{dx} y^2 = 2(y)^1 *\dfrac{dy}{dx}$
# Examples # Examples
> Differentiate $f(x) = 4\sqrt[3]{x} - \dfrac{1}{x^6}$ > Differentiate $f(x) = 4\sqrt[3]{x} - \dfrac{1}{x^6}$
2. $f(x) = 4\sqrt[3]{x} = \dfrac{1}{x^6}$ 4. $f(x) = 4\sqrt[3]{x} = \dfrac{1}{x^6}$
3. $= 4x^\frac{1}{3} - x^{-6}$ 5. $= 4x^\frac{1}{3} - x^{-6}$
4. $f'(x) = \dfrac{1}{3} * 4x^{-\frac{2}{3}} -(-6)(x^{-6-1})$ 6. $f'(x) = \dfrac{1}{3} * 4x^{-\frac{2}{3}} -(-6)(x^{-6-1})$
5. $= 4x^{-2-\frac{2}{3}} + 6x^{-7}$ 7. $= 4x^{-2-\frac{2}{3}} + 6x^{-7}$
6. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$ 8. $= \dfrac{4}{3\sqrt[3]{x^2}} + \dfrac{6}{x^7}$