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

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

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

@@ -70,13 +70,15 @@ $$ f(x) = n, \space f'(x) = nx^{n-1} $$
 # Addition/Subtraction Derivative Rule
 You can add and subtract derivatives to find what the derivative of the whole derivative would be.
 
-# Factor Derivative Rule
+# Product Rule
 $$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) - f(x)g(x)}{h} $$
 This is done by adding a value equivalent to zero to the numerator ($f(x + h)g(x) - f(x + h)g(x)$):
 $$ \dfrac{d}{dx} (f(x) * g(x)) = \lim_{h \to 0} \dfrac{f(x +h) * g(x + h) + f(x + h)g(x) - f(x+h)g(x) - f(x)g(x)}{h} $$
 
 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)*