Year 12 differentiation test to download as PDF

This Year 12 differentiation test lasts 60 minutes and is marked out of 20. It covers the whole chapter for pupils aged 16-17, and the full answers are on a separate page.

⏱️ Duration : 60 min
🎯 Total : 20 marks
❓ 5 questions

Mark/20

Question 1 – Differentiating
Differentiate (a) \(y=4x^3-\dfrac5{x}+\sqrt x\) and (b) \(y=\dfrac{x^3+2x}{x^2}\). (4 marks)
Question 2 – Tangent and normal
The curve \(y=x^3-2x^2+3\) passes through \((2,3)\). Find the equation of the tangent and the normal at this point.
Question 3 – Stationary points
\(y=2x^3-9x^2+12x\). Find the coordinates of the stationary points and determine their nature.
Question 4 – Increasing function
Show that \(f(x)=x^3+3x+1\) is increasing for all \(x\).
Find the range of values of \(k\) for which \(g(x)=x^3+kx\) is increasing for all \(x\). (Total 4 marks)
Question 5 – Problem: a fenced enclosure
A farmer has 80 m of fencing for a rectangular enclosure against a wall (three sides). The width is \(x\) m.
1. Show that the area is \(A=80x-2x^2\). (1 mark)
2. Use calculus to find the maximum area. (3 marks)

How to use this Year 12 differentiation test

First, revise with the differentiation lesson and the worksheet. Then sit the test in one go, with a timer set to 60 minutes. After that, mark your work with the differentiation test answers. If you lose marks on a question, redo it the next day without help.

Finally, you can download this Year 12 differentiation test as a PDF and print it like a real exam paper. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday.