Lesson 1: First Principles & Basic Power Rule Derivatives
Where the derivative actually comes from: the gradient of a chord, squeezed until the two points meet. Build f′(x) from the limit definition first, then earn the power rule as the shortcut it is.
🎯 Ngā Whāinga Akoranga | Learning Intentions
The limit definition of derivative f'(x) = lim_{h → 0} [f(x+h) - f(x)]/h and the power rule d/dx [x^n] = n x^(n-1) for real exponents.
Derive basic polynomial derivatives from first principles limits, apply the power rule to polynomial and fractional power functions, and find tangent gradients.
🎥 Media Anchor & Pedagogical Scaffold
Derivative as a Concept — Khan Academy
Video (7 min 16 sec, Khan Academy): An animated introduction to the derivative concept, showing the geometric interpretation of instantaneous rate of change through secant lines approaching a tangent, and introducing the limit definition and power rule. Perfect length for a full-watch classroom activity.
🧠 1. Before Viewing (Activate & Predict)
Why does finding the instantaneous rate of change require taking the limit of secant gradients as h approaches zero?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 7-minute clip. In your calculus logbook, record:
- The limit definition: Write down the formal definition f'(x) = lim_{h → 0} [f(x+h) - f(x)]/h that the video introduces.
- Geometric interpretation: Sketch a diagram showing how the secant line's slope approaches the tangent line's slope as h gets smaller.
- The power rule: Write the power rule formula d/dx [x^n] = n x^(n-1) with two worked examples from the video (e.g., for x² and x³).
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Connect tangent gradients to real-world rate of change (velocity as derivative of displacement v = ds/dt).
Immediate Task: Add to your Level 3 Calculus revision logbook (section 1): First Principles Proofs & Power Rule Practice.
⚡ Whakaoho | Do Now: The Algebra First Principles Needs (5 mins)
No calculus yet. Expand (x + h)² and (x + h)³, and simplify (5(x + h) − 5x)/h. Rewrite 1/x² and √x using index notation. Every one of these turns up inside today's limit, and this is where the marks are usually lost.
📖 Activity 1: Deriving the Power Rule From First Principles (20 mins)
In pairs, apply the limit definition f′(x) = lim(h→0) [f(x + h) − f(x)]/h to f(x) = x², then f(x) = x³. Expand, cancel the h, then let h → 0 and record what you get. Compare the two results and predict f′(x) for x⁴ before you test it. State the power rule in your own words, then check it reproduces both answers you derived. Keep the limit line visible in your working: the rule is a shortcut for it, not a replacement.
📝 Activity 2: Exam-Style Practice — Indices and Justification (15 mins)
Differentiate under exam conditions: y = 4x⁵ − 3x² + 7, y = 1/x², y = √x, y = 2x^(3/2) − 6/x. Rewrite each in index form BEFORE differentiating and show that step. For Merit, state the rule you used at each line. For Excellence, explain why the derivative of a constant is zero using the limit definition rather than by assertion.
🎫 Exit Ticket: Definition, Fluency, Meaning (5 mins)
On a slip of paper, three answers. (a) Write the limit definition of f′(x). (b) Differentiate f(x) = 4x³ − 7x + 2. (c) In one sentence: why is the derivative of a constant zero?
🏫 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: Mini-whiteboards for the limit algebra (it is worth doing wrong in public once), graphics calculator, Desmos on a projector for the chord-to-tangent animation, and the calculus logbook every ākonga keeps from today.
Pacing (57 mins): Do Now 5 · media anchor 12 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5.
Formative assessment — what to look for: The exit ticket is built to separate two things that look identical in a book of correct answers. Part (b) is fluency; part (c) is meaning. A class that gets (b) right and (c) wrong has learned a rule, not a concept — go back to the chord picture before Lesson 2, because every later rule is a shortcut for something they will not have.
Differentiation. Entry: Differentiate x², x³ and x⁴ from the limit definition, one at a time, before anyone writes a general rule. The pattern should be discovered, not announced. On level: The mixed set including negative and fractional indices, each rewritten in index form before differentiating. Extension: Prove the power rule for n = 4 from first principles using the binomial expansion of (x + h)⁴. Then say precisely where that argument stops working for n = ½ — the answer is more interesting than it looks.