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vault backup: 2025-01-30 09:48:44

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arc 2025-01-30 09:48:44 -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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8
education/math/MATH1210 (calc 1)/Derivatives.md
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@ -70,13 +70,15 @@ $$ f(x) = n, \space f'(x) = nx^{n-1} $$
# Addition/Subtraction Derivative Rule # Addition/Subtraction Derivative Rule
You can add and subtract derivatives to find what the derivative of the whole derivative would be. You can add and subtract derivatives to find what the derivative of the whole derivative would be.
# Factor Derivative Rule # Product Rule
$$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) - f(x)g(x)}{h} $$ $$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) - f(x)g(x)}{h} $$
This is done by adding a value equivalent to zero to the numerator ($f(x + h)g(x) - f(x + h)g(x)$): This is done by adding a value equivalent to zero to the numerator ($f(x + h)g(x) - f(x + h)g(x)$):
$$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) + f(x + h)g(x) - f(x+h)g(x) - f(x)g(x)}{h} $$ $$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) + f(x + h)g(x) - f(x+h)g(x) - f(x)g(x)}{h} $$
From here you can factor out $f(x + h)$ from the first two terms, and a $g(x)$ from the next two terms. From here you can factor out $f(x + h)$ from the first two terms, and a $g(x)$ from the next two terms.
Then break into two different fractions Then break into two different fractions:
$$\lim_{h \to 0} \dfrac{f(x + h)}{1} * * $$ $$\lim_{h \to 0} \dfrac{f(x + h)}{1} * \dfrac{(g(x + h) - g(x))}{h)} + \dfrac{g(x)}{1} *\dfrac{f(x + h) - f(x)}{h} $$
From here, you can take the limit of each fraction, therefore showing that to find the derivative of two values multiplied together, you can use the formula:
$$ \dfrac{d}{dx}(f(x) * g(x)) = f(x) * g'(x) + f'(x)*