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Short Study Notes β Differentiation 1
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Differentiation 1 β IB Maths AA HL: limits, derivative from first principles, standard derivatives, chain/product/quotient rules
AA HL deepens differentiation with limits, derivative from first principles, an extended derivative bank (trigonometric, exponential, logarithmic, inverse trig) and the three main rules β preparing for implicit and parametric derivatives in Differentiation 2.
What youβll learn
Mapped to the IB DP Maths AA HL subject guide (2021 onwards (applies to 2026 exams)).
AO1 β Compute limits and recognise indeterminate forms.
AO1 β State and apply the derivative from first principles.
AO1 β List standard derivatives.
AO2 β Apply chain, product and quotient rules to composite functions.
AO2 β Find tangent and normal equations.
Limits and derivative from first principles
Definition and direct evaluation.
Limit.limxβaβf(x)=L means f approaches L as x approaches a from either side.
Derivative definition. For f differentiable at x:
fβ²(x)=limhβ0βhf(x+h)βf(x)β.
Worked example. Find fβ²(x) from first principles for f(x)=x2.
Find the tangent and normal to y=x2β2x+3 at x=2.
Step-by-step solution
Step 1
y(2)=4β4+3=3. Point: (2,3).
Step 2
yβ²=2xβ2, so yβ²(2)=2. Tangent gradient =2.
Step 3
Tangent: yβ3=2(xβ2)βy=2xβ1.
Step 4
Normal gradient =β1/2. Normal: yβ3=β21β(xβ2)βy=βx/2+4.
Answer
Tangent y=2xβ1. Normal y=βx/2+4.
Key Definitions and Keywords β Differentiation 1
Definitions to memorise and the exact keywords mark schemes credit for differentiation 1 answers β sharpened from recent examiner reports for the 2026 IB DP Maths AA HL sitting.
Derivative
Examiner keyword
fβ²(x)=limhβ0β[f(x+h)βf(x)]/h β instantaneous rate of change.
Chain rule
Examiner keyword
(fβg)β²(x)=fβ²(g(x))gβ²(x).
Normal line
Examiner keyword
Line perpendicular to the tangent at the point of contact; gradient β1/fβ²(a).
Common Mistakes and Misconceptions β Differentiation 1
The traps other students keep falling into on differentiation 1 questions β taken from recent IB DP Maths AA HL examiner reports and mark schemes β and how to avoid them.
βd/dx(sin3x)=cos3x.
βΌ
Why it happens
Forgetting the chain rule.
How to avoid it
Inner derivative is 3: answer is 3cos3x.
βWriting (u/v)β²=(uvβ²βuβ²v)/v2.
βΌ
Why it happens
Sign swap.
How to avoid it
Mnemonic: 'low d high minus high d low' β (uβ²vβuvβ²)/v2.
βUsing the tangent gradient as the normal gradient.
βΌ
Why it happens
Rushing.
How to avoid it
Normal gradient =β1/tangent gradient (perpendicular).
Past paper style quiz
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Differentiation 1 β frequently asked questions
The things students keep getting wrong in this sub-topic, answered.