d/dx (x⁵) drops the power to 5x⁴ and d/dx (3x² + 4x) lands on 6x + 4, so this Year 12 differentiation sheet opens by applying the power rule term by term. The key records 6 as a gradient answer and dy/dx = 0 as a stationary condition.
Key Takeaways
d/dx(x⁵) = 5x⁴ (Q1).
d/dx(3x² + 4x) = 6x + 4 (Q2).
The key records dy/dx = 0 as a stationary condition.
Section A: Multiple Choice Questions (1 Mark Each)
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Differentiate y = x³ − 6x² + 9x with respect to x.
8.Find the gradient of y = x² − 3x at x = 2.
9.Find the equation of the tangent to the curve y = x² at the point (2, 4).
10.Find the coordinates of the stationary points of y = x³ − 3x.
11.Find the gradient of the normal to the curve y = x² − 2x at x = 3.
12.Differentiate y = 2 ÷ x with respect to x.
13.Find d²y/dx² for y = x⁴.
14.Find the set of values of x for which y = x² − 4x + 3 is an increasing function.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A particle moves in a straight line with displacement s = t³ − 6t² + 9t metres after t seconds. Find its velocity and acceleration when t = 2.
16.A rectangular field is enclosed by 120 m of fencing, so its area is A = x(60 − x) where x is the length. Find the maximum possible area of the field.
17.The number of bacteria in a culture after t hours is N = 100t³ − 900t² + 2400t. Find the rate of change of N when t = 1.
18.A ball falls a distance s = 4.9t² metres in t seconds. Find its speed after 2 seconds.
19.The profit in pounds from producing x items is P = −x² + 40x − 200. Find the number of items that maximises profit.
20.Find the x-coordinates of the stationary points of y = x³ − 3x² and state which one gives a maximum value of y.
Answer Key
1. a) 5x⁴
2. a) 6x + 4
3. a) 6
4. a) dy/dx = 0
5. a) 3x² − 1
6. a) 20
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End of Worksheet
People Also Ask
What is d/dx (x⁵)?
5x⁴, option (a) in this worksheet (Q1).
What is d/dx (3x² + 4x)?
6x + 4, option (a) in this worksheet (Q2).
What condition locates stationary points?
dy/dx = 0, from the answer key on this page.
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