Lesson 2: Differentiating Trig, Exponential & Logarithmic Functions
Radians, not degrees. Three families that are not polynomials — sin and cos, eₓ, and ln x — and a short table of derivatives you have to simply know by the external.
🎯 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 (5 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 (20 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 (15 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.
🎫 Exit Ticket: The Standard Forms (5 mins)
(a) Write d/dx[sin x] and d/dx[cos x] from memory. (b) Differentiate y = 3eₓ − 2 ln x. (c) In one sentence: why must x be in radians for d/dx[sin x] = cos x to hold?
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 alignment: Direct preparation for NCEA Level 3 Achievement Standard AS 91578, Apply differentiation methods in solving problems (external, 6 credits), against The New Zealand Curriculum (2007) Mathematics and Statistics Level 8 — Calculus. Emphasise complete algebraic working and a conclusion written in the context of the question, which is what separates Merit from Achieved.
Materials: Graphics calculators (check every one is in radian mode before the Do Now — this is the lesson where degree mode silently destroys answers), Desmos, derivative-table cards for the matching task, calculus logbook.
Pacing (54 mins): Do Now 5 · media anchor 9 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5.
Formative assessment — what to look for: Part (c) is the one that matters. A student who cannot say why radians are required will lose marks the moment a calculator is in the wrong mode, and will not know why. Check modes on the spot as you collect the slips.
Differentiation. Entry: A matching task first: function cards to derivative cards, face up, with the table visible. Fluency before recall. On level: The mixed set, with the table removed halfway through. Extension: In Desmos, graph y = aₓ for a = 2, 2.5 and 3, and find by experiment the value of a for which the gradient at x = 0 is exactly 1. Then say what you have just found.