Differentiation and Integration
Use derivatives for rates and curve behaviour and integrals for antiderivatives, signed areas, the Fundamental Theorem and simple differential equations.
How to prepare for the current ESAT
Confirm your course combination, build no-calculator fluency from Mathematics 1 upward, and use official Pearson materials for authentic timing and interface rehearsal.
Core concepts
Concept 1
Differentiate rational powers and use first and second derivatives for tangents, normals and stationary points.
Exam cue: Simplify powers before differentiating or integrating.
Concept 2
Find definite and indefinite integrals and distinguish signed integral from geometric area.
Exam cue: Use the interval and graph position when interpreting a definite integral.
Concept 3
Apply the Fundamental Theorem, integral combinations, trapezium rule and dy/dx = f(x) equations.
Exam cue: Include a constant for an indefinite integral and use conditions to determine it.
Risk pitfalls and guardrails
Treating every stationary point as a maximum or minimum without testing.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Forgetting the integration constant.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Equating a negative definite integral with a negative geometric area.
Guardrail: Do not use calculator-dependent shortcuts, import a fact outside the annual specification or overlook a unit, sign, scale or course-module boundary.
Memory anchors
Power derivative
d(xⁿ)/dx = nxⁿ⁻¹.
Stationary point
f′(x) = 0; classify with sign or second derivative.
Normal gradient
The negative reciprocal of the tangent gradient.
Power integral
∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1.
Definite integral
Evaluate the antiderivative at upper minus lower limit.
Trapezium rule
Estimate area from strip widths and endpoint ordinates.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Differentiate 5x⁴−3x²+7.
Differentiate x^(3/2).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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