GCE A-Level - Higher 1 (H1) · Mathematics (8865)

Differentiation: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Differentiation.

10 questions22 marksFree, no account

Question 1

1 mark

Find the derivative of the function \(f(x) = 3x^4 - 2\ln x + 5\) with respect to \(x\).

Question 2

1 mark

The curve \(y = \frac{1}{3}x^3 - x^2 - 3x + 5\) has two stationary points. Find the \(x\)-coordinate of the local maximum point.

Question 3

1 mark

A closed cylindrical container has a fixed volume \(V = 54\pi\) cm³. The total surface area \(S\) is given by \(S = 2\pi r^2 + 2\pi rh\), where \(r\) is the radius and \(h\) is the height. Find the value of \(r\) that minimizes the total surface area.

Question 4

1 mark

A stationary point on a curve \(y = f(x)\) occurs at \(x = c\). If \(f'(c) = 0\) and \(f''(c) > 0\), what is the nature of this stationary point?

Question 5

1 mark

Find the derivative of the function \(f(x) = \ln(3x^2 + 1)\).

Question 6

2 marks

Find the gradient of the tangent to the curve \( y = e^{2x} \) at the point where \( x = 0 \).

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Question 7

4 marks

Find the coordinates of the stationary point of the curve \( y = x^2 - 4x + 7 \) and determine its nature.

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Question 8

5 marks

Find the equation of the tangent to the curve \( y = \ln(x^2 + 1) \) at the point where \( x = 1 \).

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Question 9

2 marks

Find the derivative of the function \(f(x) = e^{2x^2 - 5x + 1}\) with respect to \(x\).

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Question 10

4 marks

The curve \(C\) has the equation \(y = 2x - e^x\).
(a) Find the \(x\)-coordinate of the stationary point on \(C\), leaving your answer in terms of natural logarithms.
(b) Determine the nature of this stationary point using the second derivative test.

Write your answer out first, then check it against the worked solution.

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