Product Rule
Let $ h(x) = f(x)g(x) $. Using first principles $$ {{ .Include "/notes/product_rule.tex" }} $$
Note that there's absolutely no reason to memorize the quotient rule for $ h(x) = f(x) / g(x) $ since $ h(x) $ is equivalent to $ f(x) g(x)^{-1} $ to which the product rule applies.
Chain Rule
Let $ h(x) = f(g(x)) $. The slope at $ x_0 $ is $$ {{ .Include "/notes/chain_rule_0.tex" }} $$
Let $ y = g(x) $ so $ y_0 = g(x_0) $. $ \lim_{x \rightarrow x_0} g(x) = g(x_0) $ or, equivalently, $ \lim_{x \rightarrow x_0} y = y_0 $ $$ {{ .Include "/notes/chain_rule_1.tex" }} $$
which can be substituted into $ \ref{chain_rule_setup} $ to give the chain rule: $$ \frac{\partial}{\partial x} f(g(x)) = f\,'(y)g\,'(x) $$ where $ y = g(x) $.
Newton's Method
Use the tangent of the function at a guess of the root to determine the next guess at the root where $ x_A $ is the initial guess and $ x_B $ is the subsequent guess
$$ {{ .Include "/notes/newtons_method_0.tex" }} $$ $$ {{ .Include "/notes/newtons_method_a.tex" }} $$The slope and intercept of the tangent is (obviously) equal at A and B, therefore subtracting $ \ref{y_B} $ from $ \ref{y_A} $ gives
$$ {{ .Include "/notes/newtons_method_1.tex" }} $$$ y_B = 0 $, $ m = f^\prime $ and $ y_A = f(x_A) $, therefore
$$ {{ .Include "/notes/newtons_method_2.tex" }} $$Solving for $ x_B $ (the next guess):
$$ {{ .Include "/notes/newtons_method_3.tex" }} $$Generalized:
$$ {{ .Include "/notes/newtons_method_4.tex" }} $$Separation Of Variables
Based on the premise that the solution is a product of functions that are each a function of just a single independent variable. eg, the heat equation with T dependent sink:
$$ {{ .Include "/notes/separation_of_variables_0.tex" }} $$Let $ u(x, t) $ be $ f(x)g(t) $, so
$$ {{ .Include "/notes/separation_of_variables_1.tex" }} $$Separate the variables
$$ {{ .Include "/notes/separation_of_variables_2.tex" }} $$The LHS equals the RHS iff they equal a constant ratio $ \lambda $
$$ {{ .Include "/notes/separation_of_variables_3.tex" }} $$