For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: derivative as tangent gradient and rate of change, standard notation, rational powers, trigonometric, exponential, and logarithmic derivatives form the main route.
School-sensitive extension: formal limit proofs and deeper first-principles work may vary by school.
2027 national comparison: K341 Topic C1 covers derivative meaning, notation, and derivatives of rational powers, sine, cosine, tangent, exponential, and logarithmic functions.
Check your school: confirm first-principles depth and the required derivative formula set.
Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 14) Differentiation Fundamentals cover? A: First principles, power rule, and basic derivative interpretations for IP AMaths.
Differentiation measures instantaneous rate of change. Know how to move from first principles to the standard rules.
Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.
The core idea is simple: Differentiation measures instantaneous rate of change.
Use it as a working check: Learn what the derivative means before memorising rules: it gives gradient, tangent slope, and rate of change.
Then go one layer deeper: Example: for y = x^2, first principles gives gradient 2x; at x = 3, the tangent gradient is 6.
Choosing the differentiation route
Before differentiating, identify what the question is asking for. The same derivative can be used as a formula, a gradient, or a rate of change.
Question cue
First move
What to finish with
"Using first principles"
Start from hf(x+h)−f(x), simplify, then let h→0.
A derivative expression, not a tangent equation.
"Differentiate" or "find dxdy"
Rewrite powers first, then apply the power rule term by term.
A simplified derivative.
"Gradient at x=a"
Differentiate first, then substitute x=a.
A number for the gradient.
"Equation of tangent"
Differentiate, find the gradient, find the point, then use y−y1=m(x−x1).
A line equation.
Common trap: substituting the point before differentiating usually gives the value of y, not the gradient. Differentiate first when the question asks about slope or rate.
Angles in these trigonometric derivative rules are measured in radians. A linear inner function adds its gradient through the chain rule, for example dxdsin(3x+1)=3cos(3x+1).
Power-rule rewrite checkpoint
Before applying the power rule, rewrite roots and reciprocals as powers of x. This makes the differentiation step mechanical.
Original form
Rewrite first
Differentiate as
Common trap
x
x21
21x−21
Keeping the power as 21 after differentiating.
x1
x−21
x21
x−2
−2x−3
kxn
Keep k outside the rule.
knxn−1
Dropping the coefficient while changing the power.
Worked check: y=x3+x22 becomes y=3x−21+2x−2, so
dxdy=−23x−23−4x−3.
Misconception to avoid: the power rule can handle fractional and negative powers, but only after the expression is written as a power of x.
2 Worked example - From first principles
Find f′(x) for f(x)=x2 using the limit definition.
