Derivative

🔑 CORE IDEA

• f′(x)f'(x)f′(x) = slope of the tangent line to f(x)f(x)f(x)

• So every point on the derivative graph tells you:
  👉 how fast f(x)f(x)f(x) is increasing or decreasing at that x

📈 1. RELATIONSHIP BETWEEN f(x)f(x)f(x) AND f′(x)f'(x)f′(x)

A. Increasing / Decreasing

• If f′(x)>0f'(x) > 0f′(x)>0 → f(x)f(x)f(x) is increasing

• If f′(x)<0f'(x) < 0f′(x)<0 → f(x)f(x)f(x) is decreasing

👉 Trick:

• Above x-axis (derivative graph) = function going up

• Below x-axis = function going down

B. Critical Points (Turning Points)

• Where f′(x)=0f'(x) = 0f′(x)=0 → horizontal tangent

• These are candidates for:

  • local max

  • local min

👉 BUT you must check sign change:

C. Concavity (Using Second Derivative Idea)

• If f′(x)f'(x)f′(x) is increasing → f′′(x)>0f''(x) > 0f′′(x)>0 → concave up

• If f′(x)f'(x)f′(x) is decreasing → f′′(x)<0f''(x) < 0f′′(x)<0 → concave down

👉 Trick:

• Look at slope of the derivative graph

  • Going up → smile 😊

  • Going down → frown ☹

D. Inflection Points

• Where concavity changes

• Happens where:

  • f′(x)f'(x)f′(x) has a local max or min

📊 2. IF YOU ARE GIVEN THE DERIVATIVE GRAPH

You can find:

✔ Where function is increasing/decreasing

→ Check sign of f′(x)f'(x)f′(x)

✔ Relative maxima/minima

→ Look at where derivative crosses x-axis

✔ Concavity

→ Look at whether derivative is rising or falling

✔ Inflection points

→ Peaks/valleys of derivative graph

📉 3. IF YOU ARE GIVEN THE ORIGINAL FUNCTION GRAPH

You can sketch the derivative:

Step-by-step:

1. Find slopes at key points

• Horizontal tangent → derivative = 0

• Steep slope → large magnitude derivative

2. Identify increasing/decreasing regions

• Increasing → derivative positive

• Decreasing → derivative negative

3. Plot rough derivative shape

🧠 HIGH-YIELD EXAM TRICKS

🔥 Trick 1: “Zeros = turning points”

• Wherever derivative = 0 → check for extrema

🔥 Trick 2: “Sign = direction”

• Positive derivative = going up

• Negative derivative = going down

🔥 Trick 3: “Slope of derivative = concavity”

• Increasing derivative → concave up

• Decreasing derivative → concave down

🔥 Trick 4: Sharp points / cusps

• Derivative does NOT exist

• Examples:

  • corners

  • vertical tangents

🔥 Trick 5: Flat but NOT turning

• If derivative = 0 but no sign change:
  → point of inflection (NOT max/min)

🔥 Trick 6: Magnitude matters

• Large ∣f′(x)∣|f'(x)|∣f′(x)∣ → steep graph

• Small ∣f′(x)∣|f'(x)|∣f′(x)∣ → flat graph

📌 4. COMMON AP/IB QUESTION TYPES

Type 1: “Where is f increasing?”

→ Look where derivative > 0

Type 2: “Find local maxima/minima”

→ Solve f′(x)=0f'(x)=0f′(x)=0 and check sign changes

Type 3: “Where is concave up/down?”

→ Look at behavior of derivative

Type 4: “Sketch f from f′”

→ Combine:

• sign

• zeros

• shape

Type 5: “Match graph of f to f′”

👉 Look for:

• turning points ↔ zeros

• steepness ↔ magnitude

📘 KEY FORMULA VISUALIZATION

f′(x)=ddxf(x)f'(x)=\frac{d}{dx}f(x)f′(x)=dxd​f(x)

⚠ COMMON MISTAKES (VERY TESTED)

• ❌ Thinking f′(x)=0f'(x)=0f′(x)=0 ALWAYS means max/min

• ❌ Ignoring sign change

• ❌ Mixing up concavity vs increasing

• ❌ Forgetting derivative doesn’t exist at sharp corners

• ❌ Assuming derivative graph is same shape as original

🧩 ADVANCED IB HL / HARD AP INSIGHT

Connecting everything:

If:

• f′(x)=0f'(x) = 0f′(x)=0 → stationary

• f′′(x)=0f''(x) = 0f′′(x)=0 → possible inflection

Then:

• You must analyze behavior around the point, NOT just value