3.2 The Derivative as a Function

Derivative functions are the mathematical superheroes of calculus. They swoop in to reveal how quickly things change, whether it's temperatures rising or populations falling. By understanding derivatives, you gain X-ray vision into function behavior.

Graphing derivatives is like decoding a secret message. The ups and downs of the original function translate into positive and negative values on the derivative graph. It's a powerful tool for spotting trends and predicting what comes next.

The Derivative Function

Definition of derivative function

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• Represents rate of change of original function $f(x)$ at any given point
  • Measures slope of tangent line to graph of $f(x)$ at specific point
  • Derivative at point $(a, f(a))$ defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, if limit exists
• Provides information about behavior of original function
  • Positive derivative values indicate original function increasing (temperature rising)
  • Negative derivative values indicate original function decreasing (population declining)
  • Zero derivative values indicate original function has horizontal tangent line or critical point (velocity momentarily zero)

Graphing derivative functions

• Graph of derivative function $f'(x)$ can be sketched based on graph of original function $f(x)$
  • Increasing intervals of $f(x)$ correspond to positive values of $f'(x)$ (car accelerating)
  • Decreasing intervals of $f(x)$ correspond to negative values of $f'(x)$ (stock price falling)
  • Critical points of $f(x)$, such as local maxima or minima, correspond to points where $f'(x) = 0$ (pendulum at highest and lowest points)
• Derivative graph provides information about concavity of original function
  • If $f'(x)$ increasing, then $f(x)$ concave up (growth rate of bacteria population)
  • If $f'(x)$ decreasing, then $f(x)$ concave down (rate of heat dissipation)
  • Points of inflection on $f(x)$ correspond to critical points of $f'(x)$ (transition from increasing to decreasing rate of change)

Continuity and Higher-Order Derivatives

Derivatives and continuity

• Function $f(x)$ differentiable at point $a$ if $f'(a)$ exists, meaning limit $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ exists
  • If function differentiable at point, it must also be continuous at that point
  • Function can be continuous at point without being differentiable there (absolute value function at $x=0$)
• Function may not be differentiable at point if:
  • Vertical tangent line (infinite slope) (tangent to $y=\sqrt[3]{x}$ at $x=0$)
  • Sharp corner or cusp ($y=|x|$ at $x=0$)
  • Discontinuity, such as jump or removable discontinuity (Heaviside step function)

Higher-order derivatives

• Derivatives of derivatives, denoted as $f''(x)$, $f'''(x)$, and so on
  • Second derivative, $f''(x)$, measures rate of change of first derivative, $f'(x)$ (acceleration)
  • Third derivative, $f'''(x)$, measures rate of change of second derivative, $f''(x)$ (jerk)
• Various applications in calculus:
  1. Second derivative helps determine concavity of function and locate points of inflection
  2. Second derivative test can be used to classify critical points as local maxima, local minima, or neither
  3. Higher-order derivatives used in Taylor series expansions to approximate functions

Applications of derivatives

• Solve optimization problems
  • Maximizing profit or minimizing cost in business scenarios (optimal production quantity)
  • Finding dimensions of container that maximize volume given constraint on surface area (cylindrical can)
  • Determining shortest path between two points (light refraction)
• Analyze rate of change in various contexts
  • Velocity and acceleration in physics (projectile motion)
  • Marginal cost, revenue, and profit in economics (diminishing returns)
  • Population growth rates in biology and demographics (logistic growth model)
• Interpret results in context of problem and communicate findings effectively
  • Relate mathematical solutions to real-world implications (optimal dosage of medication)
  • Use clear language and visuals to convey insights (graphs, tables, equations)

Differentiation Rules

• Power rule: Used for differentiating polynomial functions and functions with rational exponents
• Product rule: Applied when differentiating the product of two or more functions
• Quotient rule: Utilized for finding the derivative of one function divided by another
• Chain rule: Employed when differentiating composite functions
• Implicit differentiation: Technique used to find the derivative of a function defined implicitly rather than explicitly