Differential Calculus: Derivative Rules, Extrema, and Optimization

The Derivative and Differentiability

• Limit Definition of the Derivative at a Point:

  • $f'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a} = \lim_{h \to 0}\frac{f(a+h) - f(a)}{h}$

  • Represents the instantaneous rate of change of $f(x)$ at $x = a$ and the slope of the tangent line to $y = f(x)$ at $(a, f(a))$.

• The Derivative as a Function:

  • $f'(x) = \frac{d}{dx}f(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$

• Function Behavior and Monotonicity:

  • If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.

  • If $f'(x) < 0$ on an interval, then $f(x)$ is decreasing on that interval.

  • A function is monotonic on an interval if it is either strictly increasing or strictly decreasing across the entire interval.

• Differentiability and Continuity:

  • Differentiability implies continuity: if $f(x)$ is differentiable at $x = a$, then $f(x)$ is continuous at $x = a$.

  • The converse is false: continuity does not guarantee differentiability.

• Causes of Non-Differentiability at a Point $x = a$:

  • Corner: Left-hand and right-hand derivative limits exist but are unequal.

  • Cusp: Slopes of secant lines approach $\infty$ from one side and $-\infty$ from the other.

  • Vertical Tangent: Tangent line slope approaches $\infty$ or $-\infty$ from both sides.

  • Discontinuity: $f(x)$ has a jump, removable, or infinite discontinuity at $x = a$.

Rules of Differentiation

• Constant Rule:

  • $\frac{d}{dx}[c] = 0$

• Power Rule:

  • $\frac{d}{dx}[x^n] = n x^{n-1}$ for any real number $n$

• Constant Multiple Rule:

  • $\frac{d}{dx}[c \cdot f(x)] = c f'(x)$

• Sum and Difference Rule:

  • $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

• Exponential Rules:

  • $\frac{d}{dx}[a^x] = a^x \ln(a)$ for $a > 0$

  • $\frac{d}{dx}[e^x] = e^x$

• Product Rule:

  • $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule:

  • $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ where $g(x) \neq 0$

• Chain Rule:

  • $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$

  • Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$

  • For three composite functions: $\frac{d}{dx}[f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

Extrema and Derivative Tests

• Extrema Definitions:

  • Absolute (Global) Maximum/Minimum: $f(c) \ge f(x)$ (or $f(c) \le f(x)$) for all $x$ in the domain of $f$.

  • Local (Relative) Maximum/Minimum: $f(c) \ge f(x)$ (or $f(c) \le f(x)$) for all $x$ in an open interval containing $c$.

• Critical Points:

  • Points $c$ in the domain of $f(x)$ such that $f'(c) = 0$ or $f'(c)$ does not exist.

  • Extrema can only occur at critical points or interval endpoints.

• First Derivative Test for Local Extrema:

  • If $f'(x)$ changes sign from positive to negative at $x = c$, $f(c)$ is a local maximum.

  • If $f'(x)$ changes sign from negative to positive at $x = c$, $f(c)$ is a local minimum.

  • If $f'(x)$ does not change sign across $c$, then $f(c)$ is neither a local maximum nor a local minimum.

• Second Derivative Test for Local Extrema:

  • Assume $f'(c) = 0$ and $f''(x)$ is continuous near $c$:

  • If $f''(c) > 0$, the curve is concave up, so $f(c)$ is a local minimum.

  • If $f''(c) < 0$, the curve is concave down, so $f(c)$ is a local maximum.

  • If $f''(c) = 0$, the test is inconclusive ($f$ may have a local minimum, local maximum, or neither).

Extreme Value Theorem and Closed Interval Method

• Extreme Value Theorem (EVT):

  • If $f(x)$ is continuous on a closed interval $[a, b]$, then $f(x)$ is guaranteed to attain both an absolute maximum and an absolute minimum on $[a, b]$.

• Closed Interval Method:

  • Applied to continuous functions on $[a, b]$:

  • Step 1: Find all critical points of $f$ in the open interval $(a, b)$ and evaluate $f$ at each.

  • Step 2: Evaluate $f$ at the boundary endpoints $a$ and $b$.

  • Step 3: Compare values; the largest is the absolute maximum, and the smallest is the absolute minimum.

• Optimization on Non-Closed Domains:

  • If the domain is an open or unbounded interval, determine extrema by analyzing critical points via derivative tests, global concavity, and end-behavior limits ($x \to \pm\infty$ or domain boundaries).

Applied Optimization Framework

• Six-Step Optimization Strategy:

  • Step 1: Identify the objective quantity to be maximized or minimized.

  • Step 2: Draw a diagram and define all variables.

  • Step 3: Write a primary equation for the objective quantity.

  • Step 4: Use constraint equations to eliminate extra variables, expressing the objective formula as a function of a single variable.

  • Step 5: Identify the feasible domain based on physical and geometric limits.

  • Step 6: Find absolute extrema over the domain using the Closed Interval Method or derivative tests.

• Standard Geometric Optimization Models:

  • Cylindrical Can: Minimize total surface area $S(r) = 2\pi r^2 + \frac{256\pi}{r}$ given fixed volume $V = \pi r^2 h = 128\pi\,\text{cm}^3$ subject to height $h \ge 2\,\text{cm}$ and radius $r \ge 1\,\text{cm}$.

  • Inscribed Rectangle: Maximize area $A(x) = 2x\sqrt{4 - x^2}$ for a rectangle with base along the $x$-axis inscribed in the upper semicircle $x^2 + y^2 = 4$ of radius $2$.