9/3 Notes on Derivatives and Rate of Change: Key Concepts, Definitions, and Examples

Basic Concepts: Rate of Change, Limits, and Tangent Line

• The course starts with basic concepts: rate of change and limits, leading to the tangent line concept.

• Rate of change is the slope of the line between two points on a graph; in calculus this becomes the derivative at a point.

• Limits are used to formalize rate of change as the interval between two points shrinks to zero: you take the limit as Δx approaches zero.

• When the limit is taken at a single point, the resulting value is the instantaneous rate of change, i.e., the slope of the tangent line to the graph at that point.

• Tangent line intuition:

  • A tangent line touches the graph at exactly one point.

  • The slope of that tangent line equals the derivative at that point.

• Practical application in business: consider a cost function C(x) with various costs (material, fixed costs such as salaries, rent). Fixed costs do not vary with x, but the derivative (slope of the tangent) represents the marginal or instantaneous rate of change of cost at a production level x.

• Notation and interpretation:

  • The derivative at a = f'(a) is the instantaneous rate of change and equals the slope of the tangent line at x = a.

  • Classic notations include f'(a), dy/dx, and the derivative operator d/dx.

  • The derivative is the instantaneous rate of change or slope of the tangent line, not just an average rate.

The Derivative: Definition, Notation, and Interpretation

• Formal definition (difference quotient):
  f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

• This limit, when it exists, gives the instantaneous rate of change at x = a and the slope of the tangent line to the graph at that point.

• Slope of the tangent line and the rate of change are the same concept: derivative = slope of the tangent line.

• Quick visual: draw a small right triangle with base h and height f(a+h)-f(a); as h -> 0, the ratio height/width gives the instantaneous rate of change.

• For small h, you can use algebra (FOIL) to simplify the difference quotient before taking the limit.

Example 1: Derivative of f(x) = x^2 and f'(1)

• Given the function: f(x)=x^2. The graph passes through the origin (0,0).

• We want the slope of the tangent line at x = 1 (the rate of change at that point).

• Step 1: Use the difference quotient for a general a (then specialize to a=1):
  f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

• Step 2: Compute for f(x) = x^2:

  • f(a+h) = (a+h)^2 = a^2 + 2ah + h^2

  • f(a+h) - f(a) = (a^2 + 2ah + h^2) - a^2 = 2ah + h^2

  • \frac{f(a+h) - f(a)}{h} = \frac{2ah + h^2}{h} = 2a + h

• Step 3: Take the limit as h → 0:
  f'(a) = \lim_{h\to 0} (2a + h) = 2a.

• Special case: at x = 1, f'(1) = 2\cdot 1 = 2.

• Tangent line at a = 1:

  • f(1) = 1, f'(1) = 2

  • Equation of tangent line: y = f'(1)(x - 1) + f(1) = 2(x-1) + 1 = 2x - 1.

• Relationship recap:

  • Slope of tangent line at x = a is f'(a).

  • Tangent line equation can be written as y = f'(a)(x - a) + f(a).

Example 2: Constant function and derivative of a horizontal line

• If the function is constant, e.g., f(x)=c, then the derivative is zero for all x:
  f'(x) = 0.

• Example given: if a function has a constant value like f(x)=60, then \frac{d}{dx}f(x)=0.

Example 3: Continuity, differentiability, and non-differentiable points

• Key idea: differentiability at a point implies continuity at that point; if a function is not continuous at a, the derivative does not exist at a.

• Points raised in the transcript:

  • At x = 1, the function is not continuous due to a jump; derivative does not exist there.

  • At x = 3, the function is not defined, so it is not continuous there either; derivative does not exist.

• Concept: Continuity requires that both left-hand and right-hand limits exist and equal the function value at the point:

  • If \lim{x\to a^-} f(x) = \lim{x\to a^+} f(x) = f(a), then f is continuous at a.

  • If continuity fails, derivative does not exist at that point.

• Common sources of non-differentiability:

  • Point of discontinuity (jump, infinite discontinuity, etc.),

  • Corner or cusp (sharp turn),

  • Vertical tangent (infinite slope) — may or may not be differentiable depending on the context.

Theorem-like ideas: Notation, interpretive points, and the power of the derivative

• Notation recap:

  • f'(x) denotes the derivative function when it exists, with f'(a) giving the slope at x = a.

  • dy/dx is another way to denote the derivative with respect to x.

• Tangent line basics remain the same: the derivative gives the slope of the tangent to the graph at that point.

• The graph of a constant function is a horizontal line; its slope is 0.

Example 4: The power rule and derivative rules for simple polynomials

• Power rule (concept): If f(x) = x^n, then
  \frac{d}{dx} x^n = n x^{n-1}.

• Examples:

  • For n = 7: \frac{d}{dx} x^7 = 7 x^{6}.

  • For negative exponent, e.g., x^{-2}: \frac{d}{dx} x^{-2} = -2 x^{-3}.

• The transcript notes that derivative results can involve integers, fractions, or decimals depending on the problem; the same rules apply.

• Also mentioned: coefficients multiply through in linear terms; e.g., if f(x) = ax^n, then f'(x) = a n x^{n-1}.

Example 5: Using FOIL and expansion to find the derivative outline

• Another common method to derive the derivative of a polynomial is to expand and simplify:

  • Consider f(x) = (x+h)^2; expand: $(x+h)^2 = x^2 + 2xh + h^2$.

  • Then compute the difference quotient: \frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h.

  • Take the limit as h → 0: \frac{d}{dx} x^2 = 2x.

• This demonstrates both FOIL and simplification routes to the derivative.

Example 6: A quadratic with mixed terms and a quick derivative

• A common example mentioned: f(x) = -0.02 x^2 + 21x.

• Derivative (by applying the power rule to each term):
  f'(x) = -0.04x + 21.

• You can evaluate at a = 0, 1, or any x as needed: e.g., at x = a, the instantaneous rate of change is f'(a) = -0.04a + 21.

• The transcript suggests recognizing that problem difficulty varies and that recognizing derivative rules speeds up solving.

The rule set: Three quick reminders for differentiability and differentiation practice

• Third major takeaway: differentiability implies continuity; lack of continuity implies non-differentiability.

• For a constant function, derivative is zero; the slope of the horizontal line is 0.

• The derivative measures the instantaneous rate of change, not just an average change.

• The derivative of a sum is the sum of derivatives, and the derivative of a product requires product rule (not covered in depth here, but arises naturally from the rules above).

• Quick notational check:

  • The derivative is denoted as f'(x) or as \dfrac{dy}{dx}; the function value is still f(x) or y.

Practical tips for learning and practice

• The material emphasizes building from foundations: limit, rate of change, tangent line, and derivative as the instantaneous rate of change.

• The progression moves from conceptual understanding to algebraic manipulation and then to applying rules (like the power rule) for quicker computation.

• When faced with difficult problems, break them down into known rules and apply algebraic simplification (e.g., expand, factor, or FOIL) before taking limits.

• Real-world relevance: derivatives model marginal changes in cost, revenue, or other business quantities; the slope of the tangent line at a point provides insight into how the quantity changes right at that production level.

• Study strategy highlighted in the transcript:

  • Practice daily in short bursts (roughly 20 minutes per day) rather than long, infrequent sessions.

  • Build fluency with basic rules so that you can recognize patterns quickly in more difficult problems.

  • Ensure you can simplify expressions to their simplest forms, especially in multiple-choice contexts where the correct choice is the simplest form.

Quick recap: Key equations to memorize

• Difference quotient (derivative definition):
  f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}

• Power rule (for a general x^n):
  \frac{d}{dx} x^n = n x^{n-1}

• Example for f(x)=x^2:

  • f'(a) = \lim{h\to 0} \frac{(a+h)^2 - a^2}{h} = \lim{h\to 0} (2a + h) = 2a

  • At a = 1: f'(1) = 2, tangent line: y = 2(x-1) + 1 = 2x - 1

• Derivative of a constant:
  \frac{d}{dx} c = 0

• Derivative of a linear function f(x) = ax + b:

  • f'(x) = a

• Derivative of a quadratic with mixed terms: if f(x) = -0.02 x^2 + 21 x, then

  • f'(x) = -0.04 x + 21

• Non-differentiable conditions: jumps (discontinuities), corners, cusps, or undefined points; derivative does not exist at those points.

Note: Mirrors the classroom flow described in the transcript

• The teacher connects abstract limits to concrete computations (FOIL, expanding polynomials) to build intuition.

• Emphasizes the learning curve: from initial difficulty to manageable, routine calculations with practice.

• Encourages consistent practice and familiarity with simplification to facilitate test-taking, especially for multiple-choice formats.