Calc-1 Discussion 04 | Tony's Intertwingled Memex

Discussion Topics covered: Rates of change and visual differentiability, computing derivatives from first principles via limit definition, power rule family, algebraic pre-processing and lowest-power post-processing, product rule, quotient rule, trigonometric derivatives, higher-order derivatives and tangent line linearization, logarithmic and exponential properties, decomposing composite functions, isolating target variables

Part A: Review This Week's Lecture Topics (~30 mins)

Rates of Change & Visual Differentiability

Distinguishing average rates of change (secant line slopes $\frac{\Delta y}{\Delta x} = \frac{f(a+h)-f(a)}{h}$ ) from instantaneous rates of change (tangent line slopes $f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ )
Calculating derivatives on piecewise linear segments directly using the constant segment slope $\frac{y_2 - y_1}{x_2 - x_1}$
Diagnosing the four graphical failure modes of differentiability on coordinate curves: sharp corners, cusps, vertical tangents, and discontinuities
Proving differentiability implies continuity ( $\text{differentiable} \implies \text{continuous}$ ), and explaining why continuity is a necessary but not sufficient condition for differentiability (e.g., at sharp corners and cusps)

Computing Derivatives from Scratch via Limit Definition

Applying the limit definition of the difference quotient $f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ from first principles to resolve $\frac{0}{0}$ indeterminate forms
Expanding polynomial difference quotients via binomial expansion ( $(x+h)^2, (x+h)^3$ ) and factoring out $h$ to cancel the denominator
Combining rational difference quotients over common denominators to clear compound fractions and isolate $h$
Rationalizing radical difference quotients using radical conjugates $(\sqrt{A} + \sqrt{B})$ and the difference of squares identity $(a-b)(a+b) = a^2 - b^2$ to release $h$ from under radicals

Fundamental Differentiation Rules & Exponent Pre/Post-Processing

Differentiating polynomials term-by-term using the Constant Rule ( $\frac{d}{dx}[c] = 0$ ), Linear Identity Rule ( $\frac{d}{dx}[x] = 1$ ), Constant Multiple Rule ( $\frac{d}{dx}[c f(x)] = c f'(x)$ ), Sum/Difference Rules, and Power Rule ( $\frac{d}{dx}[x^n] = n x^{n-1}$ )
Pre-processing expressions prior to differentiation: converting radicals to fractional powers ( $\sqrt[n]{x^m} \to x^{\frac{m}{n}}$ ) and lifting denominators to negative powers ( $\frac{c}{x^n} \to c x^{-n}$ ) to bypass cumbersome quotient calculations
Avoiding the constant denominator trap by factoring out scalar denominators as constant coefficients (e.g., $\frac{f(x)}{k} = \frac{1}{k}f(x)$ ) rather than executing the Quotient Rule
Post-processing complex derivative results by factoring out the lowest (most negative) powers to yield clean factored expressions

Product, Quotient, and Trigonometric Derivative Rules

Applying the Product Rule $\frac{d}{dx}[uv] = u'v + uv'$ to multi-factor functions without committing the naive term-by-term multiplication error ( $\frac{d}{dx}[uv] \neq u'v'$ )
Applying the Quotient Rule $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$ ("Low dHigh minus High dLow over Low Low") and accurately distributing negative signs across numerator polynomial terms
Evaluating derivatives of all six fundamental trigonometric functions: $\frac{d}{dx}[\sin x] = \cos x, \qquad \frac{d}{dx}[\cos x] = -\sin x, \qquad \frac{d}{dx}[\tan x] = \sec^2 x$ $\frac{d}{dx}[\csc x] = -\csc x \cot x, \qquad \frac{d}{dx}[\sec x] = \sec x \tan x, \qquad \frac{d}{dx}[\cot x] = -\csc^2 x$

Higher-Order Derivatives & Tangent Line Linearization

Computing successive higher-order derivatives ( $f''(x), f'''(x), \dots, f^{(n)}(x)$ ), exploiting polynomial termination when derivative order exceeds polynomial degree ( $k > n \implies f^{(k)} = 0$ ) and 4-step cyclic trigonometric periodicity ( $f^{(n)}(x) = f^{(n \bmod 4)}(x)$ )
Constructing tangent line equations using point-slope form $y - f(a) = f'(a)(x-a)$ from function formulas and discrete tabular sensor logs
Formulating linear approximations (linearizations) $L(x) = f(a) + f'(a)(x-a)$ for local physical system estimation

Part B: Next Week's Prerequisite Refresher (~30 mins)

Review the following prerequisite skills:

Part C: General Q/A

Open the floor to questions from students regarding any topic