Estimate small changes with differentials
Use dy ≈ f′(x)dx to estimate a small output change and assess when a local linear approximation is useful.
Section 14 · Calculus
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which derivative belongs in the estimate dy ≈ ?
What you will learn
- Use the derivative as a local linear approximation
- Estimate a small change in a function value
- Compare an estimated change with an exact change when available
A tangent line estimates a nearby function value
For a small change dx near x, the corresponding change in y is approximately dy = f′(x)dx. Thus f(x + dx) ≈ f(x) + f′(x)dx.
The estimate uses the tangent gradient at the starting point. It is usually more accurate when dx is small and the curve does not bend sharply over that interval.
Approximation and exact change are different
The exact change is Δy = f(x + dx) − f(x). The differential dy = f′(x)dx is a linear estimate of that change, not necessarily exactly equal to it.
Keep units and signs: a negative dx estimates a change in the opposite input direction. A positive derivative with a negative dx gives a negative estimated dy.
Pause and explain
If f′(3) = 4 and dx = 0.02, what is the estimated dy?
Worked example
Use differentials to estimate the change in x³ when x increases from 2 to 2.01. Compare with the exact change.
Show the worked solution
- For f(x) = x³, f′(x) = 3x²; at x = 2, f′(2) = 12.
- With dx = 0.01, dy ≈ 12 × 0.01 = 0.12.
- The exact change is 2.01³ − 2³ = 0.120601, close to the estimate for this small step.
Answer Estimated change 0.12; exact change 0.120601.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Estimate the change in √x when x increases from 9 to 9.06.
Give me a hint
Use f′(x) = 1/(2√x) at x = 9.
Compare my reasoning
- f′(9) = 1/6.
- dx = 0.06.
- dy ≈ (1/6)(0.06) = 0.01.
The estimated change is 0.01.
Look for these in your work
- Derivative evaluated at starting point
- Small input difference found
- Approximate change calculated
Estimate the area change of a circle when its radius increases from 5 m to 5.02 m.
Give me a hint
Use A = πr² and dA ≈ 2πr dr at r = 5.
Compare my reasoning
- dr = 0.02 m.
- dA ≈ 2π(5)(0.02).
- Evaluate the approximate area change.
dA ≈ 0.2π m² (about 0.628 m²).
Look for these in your work
- Radius change identified
- Derivative evaluated locally
- Area units squared
Estimate √104 using a tangent approximation near x = 100.
Give me a hint
Let f(x) = √x, use dx = 4 and f′(100) = 1/20.
Compare my reasoning
- f(100) = 10.
- dy ≈ (1/20)(4) = 0.2.
- Add the estimated change to 10.
√104 ≈ 10.2.
Look for these in your work
- Starting value selected
- Differential estimate calculated
- Approximation stated
Common mistakes
- Using f(x) instead of f′(x) for the gradient
- Treating an approximation as exact
- Using a large input change without considering curve shape
- Ignoring a negative sign in dx
What can you explain now?
What expression estimates the small change in y = f(x) for input change dx?
Compare with the explanation
dy ≈ f′(x)dx
Multiply the local gradient by the small input change.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How does a differential estimate differ from an exact finite change?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.