Differential Calculus is one of 10 topics in the Grade 12 Mathematics CAPS syllabus, covered across 3 lessons on Study Vault. Below are real practice questions from this topic with full worked answers.
Sample practice questions
Easy
1. The definition of the derivative by first principles is f′(x)=limh→0 of:
[f(x+h) - f(x)] / h
[f(x) - f(h)] / x
f(x+h) - f(x)
f(x) / h
Answer: [f(x+h) - f(x)] / h — This is the formal definition of the derivative: the limit of the average rate of change as the interval h shrinks to 0.
Medium
2. For f(x)=x2, find f(x+h)−f(x) by expanding (x+h)2−x2. Give the simplified result in terms of x and h, in the form 2xh+h2.
Answer: 2xh+h^2 — (x+h)2−x2=x2+2xh+h2−x2=2xh+h2.
Hard
3. Continue from f(x)=x2: divide 2xh+h2 by h to get 2x+h, then let h tend to 0. What is f′(x)?
Answer: 2x — As h tends to 0, 2x+h tends to 2x, so f′(x)=2x — matching the power rule result.
