HKDSE Maths | Applications of Differentiation: Teacher Prep Guide

HKDSE Maths | Applications of Differentiation: Teacher Prep Guide | Tangents, Extrema, Graphs

2026-11-02

Tutor lesson-prep guides · Secondary Maths

Students can differentiate but cannot use it. For f(x)=x cubed minus 6x squared plus 9x plus 1 the stationary points sit at x=1 and x=3, giving a maximum of 5 and a minimum of 1. One lesson drills four steps: differentiate, solve, classify, answer.

Objectives — Students can: (1) solve f prime of x equals zero for every stationary point and classify it with the second derivative; (2) write the tangent at a given point and tidy it into the form y equals mx plus c; (3) state where the curve rises and falls from the sign of the derivative, naming the turning points.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Differentiation rules	Term by term, constants vanish	x cubed becomes 3x squared
Solving quadratics	Factorise before using the formula	Set the derivative to zero
Tangent idea	Slope equals the derivative at that point	Slope equals f prime of a

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-8 min	Diagnostic	Three pure differentiation items, done alone	All three correct to continue
8-20 min	Stationary points	Solve the derivative, substitute back for y	Both points found
20-32 min	Classification	Second derivative at both points	Maximum and minimum correct
32-44 min	Tangent equation	At x=2, tidy to y=mx+c	Slope -3
44-54 min	Graph reading	Signs of the derivative give the intervals	All three intervals
54-60 min	Exit check	Three quick questions	All three correct

03 Teaching script (read it out)

Scene 1: A stationary point is a candidate, not an answer

The derivative is 3(x-1)(x-3), so the stationary points are x=1 and x=3.
Substituting back gives f(1)=5 and f(3)=1.
The second derivative is 6x-12: at x=1 it equals -6, so that point is a maximum; at x=3 it equals 6, so that point is a minimum.
Rule: every stationary point is written with its classification, never the x value alone.

Ask the student: why does a zero derivative not guarantee an extremum?

Scene 2: A tangent needs three things: point, slope, equation

At x=2, f(2)=3 and the slope is -3.
Point-slope form gives y minus 3 equals -3 times x minus 2, which tidies to y=-3x+9.
Two common slips: writing the slope alone as the answer, and giving the normal instead of the tangent.

Ask the student: which three pieces of information must a tangent answer contain?

Scene 3: The sign of the derivative is the graph story

The derivative is positive left of 1, negative between 1 and 3, and positive again to the right of 3.
So the curve rises, falls, then rises: up, down, up.
Answer with a statement and a reason: it is increasing here because the derivative is positive.

Ask the student: what does the derivative changing from positive to negative tell you?

04 Question types and worked steps

Type	Example	Steps	Answer
Calculation	Find stationary points	Set the derivative to zero and factorise	x=1 and 3
Calculation	Classify extrema	Substitute into the second derivative	Negative means maximum
Calculation	Tangent equation	Find point and slope, then tidy	y=-3x+9
Explanation	Increasing and decreasing	Read derivative signs, write intervals	Three intervals

Sanity check (teach this): Reverse-check three ways: the stationary points must satisfy the derivative; the tangent and the curve must agree at x=2 with y=3; the second derivative sign must match an up-down-up shape.

05 Independent practice (3 questions)

Question	Answer	What to watch
Find the stationary points and extrema of the cubic	Differentiate, solve, classify	Maximum 5, minimum 1
Write the tangent at x=2	Point-slope then tidy	y=-3x+9
State the intervals of increase and decrease	Use derivative signs	Three intervals
Explain why the point at x=1 is a maximum	Use the second derivative	Mentions a negative value

06 Common mistakes and what to say

Mistake	Student writes	Say this
Stopping once the derivative is zero	Classification marks lost	“The stationary point is only a candidate.”
Using the derivative value as the y coordinate	Whole tangent line wrong	“The point uses the function, not the derivative.”
Leaving the tangent in point-slope form	Format marks lost	“Point-slope is the middle step.”
Saying it rises and falls with no reason	No marks on explanation items	“One conclusion, one derivative sign.”

07 Homework (paper, section and question numbers)

Three stationary point and extremum items — Practice paper questions 1 to 3
Two tangent equations — Practice paper questions 4 and 5
One graph explanation — Practice paper question 6

08 Exit check (last 5 minutes)

Question	Answer	Decision
What comes after the derivative equals zero?	Classify with the second derivative	Or check the sign change
What is the tangent slope at x=2?	-3	Not 3
What does positive to negative mean?	A maximum point	Not a minimum

09 Tutor FAQ

They differentiate fine but freeze on applications, what helps?

Four fixed steps written on one line each: differentiate, solve, classify, answer.

When should they use the formula for quadratics?

Factorise first; the formula only when factors are not obvious.

Should normals be taught at the same time?

Separate lessons; normals are one line about negative reciprocals.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.