For y = sin x the derivative dy/dx is cos x and for y = eˣ it stays eˣ, so this IB sheet opens with the two derivatives students reuse everywhere. A product involving x² sin x gives 2x sin x + x² cos x in the key.
Key Takeaways
dy/dx = cos x when y = sin x (Q1).
dy/dx = eˣ when y = eˣ (Q2).
The key records 2x sin x + x² cos x for a product rule answer.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.What is dy/dx if y = sin x?
(a) cos x(b) −cos x(c) −sin x(d) tan x
2.What is dy/dx if y = eˣ?
(a) xeˣ(b) eˣ(c) 1 ÷ eˣ(d) ln x
3.What is dy/dx if y = ln x?
(a) 1/x(b) ln x(c) x(d) −1/x
4.What is dy/dx if y = x² sin x?
(a) 2x sin x + x² cos x(b) 2x cos x(c) 2x sin x(d) x² cos x
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Find dy/dx if y = sin 3x.
8.Find dy/dx if y = e³ˣ.
9.Find dy/dx if y = x²eˣ.
10.Find dy/dx if y = (x² + 1) ÷ x.
11.Find the stationary point of y = x² − 4x + 1 and state whether it is a maximum or a minimum.
12.Find dy/dx if y = tan x.
13.Find the gradient of y = x³ at x = 2.
14.Find the second derivative of y = 2x³ − 3x².
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.Revenue is R(x) = 50x − x². Find the number of units x that maximises revenue.
16.A farmer uses 100 m of fence for a rectangle against a wall. If the width is x, the area is A = x(50 − x). Find the width x that maximises the area.
17.A firm's cost is C(x) = 2000 + 10x + 0.05x² dollars. Find the marginal cost at x = 100.
18.A particle's speed is v(t) = 3t² − 12t + 9. Find the time when the acceleration is zero.
19.A population is P(t) = 1000 × e to the 0.05t. Find the rate of growth at t = 10.
20.A sphere's radius grows at 2 cm/s. Find dV/dt when r = 5, where V = (4 ÷ 3)πr³.
Answer Key
1. a) cos x
2. b) eˣ
3. a) 1/x
4. a) 2x sin x + x² cos x
5. a) 6x
6. a) 6(2x + 1)²
7. Refer to solution guide.
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20. Refer to solution guide.
End of Worksheet
People Also Ask
What is dy/dx when y = sin x?
cos x, option (a) in this worksheet (Q1).
What is dy/dx when y = eˣ?
eˣ, option (b) in this worksheet (Q2).
What is the derivative of x² sin x?
2x sin x + x² cos x, from the answer key on this page.
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