3.2.2 The Derivative as a Function

Understanding the Graph of a Derivative

• The derivative f' of a function f represents the slope (rate of change) of f at every x–value.
• Sketching f' from a given graph of f requires answering three repetitive questions for each segment:
  • What is the numerical slope on that segment?
  • Is the slope constant, increasing, or decreasing?
  • Does the slope change sign (positive/negative) or become 0 anywhere?
• Key visual cues on the original graph:
  • Horizontal tangents → f'=0.
  • Increasing portions of f → f' > 0 (graph of f' lies above the x–axis).
  • Decreasing portions of f → f' < 0 (graph of f' lies below the x–axis).
  • Sharp corners, cusps, jumps, or vertical tangents → f' is undefined (open circles or breaks on the f' graph).

Example 1 – Piecewise Linear Function

• Original f is composed of three straight-line pieces.
• Segment 1: same slope from x=a to x=b.
  • Numerical value given: m=-1\Rightarrow f'=-1 on (a,b).
  • Depicted on f' as a horizontal line at y=-1.
• At x=b there is a sharp corner → f' undefined ⇒ open circle on f'.
• Segment 2: slope m=+1 on (b,c).
  • Horizontal line at y=1 for f'.
• At x=c another sharp corner → open circle on f'.
• Segment 3: slope m=-\tfrac12 on (c,d).
  • Horizontal line at y=-\tfrac12 until the graph ends at d.
• Because every piece of f is linear, f' is piecewise constant.

Example 2 – Smooth Curve with Turning Points (Graph labeled g)

• The graph shows three visible turning points at x=-3,-1,1.
• Slope analysis:
  • At turning points → g'(x)=0 (place points on x-axis of new graph).
  • Region 1: x
  • Region 2: -3<x<-1 → slope <0, goes from 0 down then back up to 0.
  • Region 3: -1
  • Region 4: x>1 → slope <0, starts at 0 and becomes more negative.
• Resulting g' is a qualitative sketch: four connected lobes alternating above and below the x–axis, touching it at x=-3,-1,1.
• Practical reminder: while drawing g', do not be tricked by the concavity or by the slope of the sketched derivative itself—the location (positive/negative) is dictated solely by the slope of g.

Example 3 – Curve With Multiple Behaviors

• Steps to sketch f':
  • Mark all points where f' = 0 (horizontal tangents). One such point is around x \approx a.
  • Identify sign of slope on each interval:
  • Left of the zero point → slope <0.
  • Immediately right of the zero point → slope >0 (increases quickly).
  • Far right branch and far left branch both have slope <0 and approach 0 asymptotically.
  • Translate that sign information into regions above or below the axis for f'.
• The resulting sketch shows three lobes, two below the axis (negative slope) sandwiching one above (positive slope).

Relationship Between Differentiability and Continuity

• Fundamental theorem: “If f is differentiable at a, then f is continuous at a.”
  • Contrapositive: “If f is not continuous at a, then f is NOT differentiable at a.”
• Proof outline given in textbook (limits definition of derivative → limit definition of continuity); video simply states and applies it.

When Does a Derivative Fail to Exist? (Three Classical Cases)

• 1. Discontinuity: any form of jump, removable, or infinite discontinuity ⇒ f' undefined.
• 2. Corner/Cusp: left-hand slope \neq right-hand slope, so limit of the difference quotient fails.
• 3. Vertical Tangent: slope approaches \pm\infty ⇒ derivative undefined (does not produce a finite number).

Worked Example – Function g on [-4,4]

• Problem (a): Find x where g is NOT continuous.
  • Observations: jump at x=-2 plus hole at x=2 → not continuous at x=-2, 2.
• Problem (b): Find x where g is NOT differentiable.
  • Non-continuous points automatically included: x=-2,2.
  • Add sharp corner at x=0.
  • Therefore g' does not exist at x=-2,0,2.
• Problem (c): Sketch g'.
  • Start by marking where g' = 0 (horizontal tangents): at x=-2 (open), x=2 (open).
  • Interval [-4,-2): original segment is linear, estimated slope \approx 1 → draw constant line y=1 with open circle at x=-2.
  • Interval (-2,0): original slope negative and changing (curved); graph g' in negative region, connecting to origin at x=0.
  • Interval (0,2): slope positive, decreasing to 0 → graph g' above axis, open circle at x=2.
  • Interval (2,4]: slope negative constant (linear piece) → graph g' as horizontal line below the axis.
• Overall sketch shows four separate branches, undefined at x=-2,0,2.

Ethical & Pedagogical Notes

• When communicating slopes/derivatives to learners, emphasize:
  • Distinction between the slope of f and the slope of the picture you are currently drawing (which is f'!).
  • Open circles must be used to highlight non-existence of derivatives.
  • “Sketch” means shapes are qualitative; exact heights depend on measured slopes when numeric information is available.

Key Numerical & Symbolic Facts

• Slope formula for straight segment: m=\dfrac{\Delta y}{\Delta x}.
• Constant slopes in Example 1: m1=-1,\; m2=1,\; m_3=-\dfrac12 → horizontal lines on f'.
• Theorem statement (precise form):
  \text{If }\; f \text{ is differentiable at } a, \text{ then }\; \lim_{x\to a} f(x)=f(a) \Rightarrow f \text{ is continuous at } a.

Study Tips & Connections

• Always label critical x–values (turning points, corners, discontinuities) before drawing f'.
• Check sign changes: if f switches from increasing to decreasing, f' crosses the x–axis from positive to negative.
• Recall previous lectures on limits; differentiability ultimately involves a two-sided limit of the difference quotient.
• Real-world relevance: determining speed from a position–time graph, detecting acceleration sign changes from a velocity graph, etc.
• Practice: pause before viewing instructor’s sketch, attempt your own, then compare; self-explanation enhances retention.