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: $x<-3$ → slope $>0$ but trending toward $0$.
  • Region 2: -3<x<-1 → slope <0, goes from $0$ down then back up to $0$.
  • Region 3: $-1<x<1$ → slope $>0$, symmetric rise and fall, stays above axis.
  • 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: $m<em>1=-1,\; m</em>2=1,\; m_3=-\dfrac12$ → 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.