Lesson 2: Differentiating Trig, Exponential & Logarithmic Functions
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Derivative formulas for sin(x), cos(x), tan(x), e^x, and ln(x), including constant scalar multipliers.
Differentiate trigonometric, exponential, and natural logarithm functions, and calculate gradient values at specific angles (radians) and domain values.
🎥 Media Anchor & Pedagogical Scaffold
Derivatives of sin(x) and cos(x) — Khan Academy
Video (3 min 41 sec, Khan Academy): A focused, visual introduction to differentiating trigonometric functions, showing why d/dx [sin(x)] = cos(x) and d/dx [cos(x)] = -sin(x) using the unit circle and limit concepts. Perfect for a quick, focused lesson segment.
🧠 1. Before Viewing (Activate & Predict)
Why is the natural exponential function e^x unique in calculus, being equal to its own derivative?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 3¾-minute clip. In your calculus logbook, record:
- Sine derivative: Write d/dx [sin(x)] = cos(x) and sketch the unit-circle reasoning the video shows.
- Cosine derivative: Write d/dx [cos(x)] = -sin(x) and note why there's a negative sign.
- Worked example: If f(x) = 3 sin(x), calculate f'(π/4) using the derivative formula. (Answer: 3 cos(π/4) = 3√2/2.)
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Remind students that all trigonometric calculus MUST be evaluated in RADIANS, never degrees (common NCEA exam error).
Immediate Task: Add to your Level 3 Calculus revision logbook (section 2): Transcendental Function Derivatives Problem Set.
⚡ Whakaoho | Do Now: Radians, e and ln (10 mins)
Convert 30°, 90° and 180° to radians. Write the exact values of sin(π/6) and cos(π/3). Simplify ln(e³) and e^(ln 5). Today's derivatives are only valid in radians, which is the single most common source of lost marks in this topic.
📖 Activity 1: Building the Standard Derivative Table (25 mins)
Work in pairs to build the table for sin x, cos x, tan x, eˣ and ln x. Test each one numerically before you accept it: for f(x) = sin x at x = 1, compute [sin(1 + 0.001) − sin(1)]/0.001 and compare it with cos(1). Do the same check for eˣ at x = 0. Then explain to your partner why eˣ is the only function that is its own derivative, and what that means about its graph's gradient.
📝 Activity 2: Exam-Style Practice — Mixed Standard Forms (20 mins)
Differentiate y = 3sin x − 2cos x, y = 5eˣ + ln x, and y = 4 tan x. For Merit, keep the coefficient handling explicit. For Excellence, find the gradient of y = eˣ at x = 0 and explain what that tells you about the tangent line there, referring to the numerical check you ran.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 Alignment: Direct preparation for NCEA Level 3 Calculus — External (Apply differentiation methods in solving problems). Emphasise complete algebraic working and clear context conclusions for Merit/Excellence grades.