Topic Review — Further Differentiation

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1. Fluency

   Q1 — Chain rule: identify and differentiate

   Differentiate each function using the chain rule.

   (a) y = (3x + 1)⁵   (b) y = (x² − 4)³   (c) y = √(2x + 7)   (d) y = 1/(x + 3)²

2. Fluency

   Q2 — Product rule: basic applications

   Differentiate each function using the product rule.

   (a) y = x(x + 5)   (b) y = x²(2x − 3)   (c) y = (x + 1)(x² + 4)   (d) y = x³√x

3. Fluency

   Q3 — Quotient rule: basic applications

   Differentiate each function using the quotient rule.

   (a) y = (x + 3)/(x − 1)   (b) y = x²/(x + 2)   (c) y = (2x + 1)/x²   (d) y = 1/(x² + 4)

4. Understanding

   Q4 — Chain rule: gradient at a point

   For each curve, find the gradient at the specified point.

   (a) y = (2x − 1)&sup4; at x = 1   (b) y = (x² + 2)³ at x = 1   (c) y = √(3x + 1) at x = 1   (d) y = (1 − 2x)³ at x = 0

5. Understanding

   Q5 — Product rule: equation of tangent

   Find the equation of the tangent to each curve at the given point.

   (a) y = x(x + 4) at x = 2   (b) y = x²(x − 1) at x = 2   (c) y = (x + 1)(x² − 2) at x = 1

6. Understanding

   Q6 — Quotient rule: turning points

   Find all x-values where dy/dx = 0 for each function.

   (a) y = x/(x² + 3)   (b) y = x²/(x + 2)   (c) y = (x − 2)/(x² + 1)

7. Understanding

   Q7 — Choosing the correct rule

   State which rule (chain, product, quotient, or power rule alone) you would use first, and then differentiate each function.

   (a) y = (5x − 2)³   (b) y = x(x + 1)⁴   (c) y = (x + 1)/(x² + x)   (d) y = 3x⁴ − 7x² + 2

8. Problem Solving

   Q8 — Chain rule: stationary points and nature

   For the curve y = (x² − 4)⁴:

   (a) Find dy/dx.   (b) Find all stationary points.   (c) Classify each stationary point using a sign diagram.   (d) Find the coordinates of each.

9. Problem Solving

   Q9 — Product rule combined with chain rule

   Find dy/dx for each function. Factorise your answers fully.

   (a) y = x²(x + 1)⁵   (b) y = (x − 1)(2x + 3)&sup4;   (c) y = x(x² − 1)³

10. Problem Solving

    Q10 — Quotient rule: nature of stationary points

    For y = (x² + 1)/(x):

    (a) Find dy/dx using the quotient rule.   (b) Find all stationary points.   (c) Determine the nature of each.   (d) Verify by rewriting y = x + 1/x and differentiating using the power rule.

11. Problem Solving

    Q11 — Motion: chain rule application

    A particle moves along a line. Its displacement at time t ≥ 0 is s(t) = (t² + 1)³ metres.

    (a) Find the velocity v = ds/dt.   (b) Find the acceleration a = dv/dt.   (c) Find the velocity when t = 1.   (d) Is the particle ever stationary (v = 0)? Justify.

12. Problem Solving

    Q12 — Applied optimisation using quotient rule

    A function modelling efficiency is E(x) = x/(x² + 1) for x ≥ 0.

    (a) Find E′(x).   (b) Find the value of x that maximises E(x).   (c) Find the maximum value of E.   (d) Confirm it is a maximum using a sign diagram or second derivative argument.

13. Problem Solving

    Q13 — Full mixed: identify rule, differentiate, apply

    For each function, state the rule(s) required and find dy/dx. Then find the gradient at x = 1.

    (a) y = x²(3x − 1)³   (b) y = (x + 1)² / (x² + 1)   (c) y = (x² − 1)(x² + 1)⁴

14. Problem Solving

    Q14 — Rate of change application

    A population of bacteria at time t hours is modelled by P(t) = 100t / (t² + 9) (hundreds of bacteria).

    (a) Find P′(t).   (b) Find the time at which the population peaks.   (c) What is the maximum population?   (d) Find the rate of change of population at t = 6. Is the population increasing or decreasing?

15. Problem Solving

    Q15 — Extended: all three rules

    Consider the function f(x) = x(x + 1)² / (x² + 1).

    (a) Find the numerator's derivative using the product and chain rule.   (b) Apply the quotient rule to find f′(x).   (c) Evaluate f′(0).   (d) Determine whether f has a stationary point at x = 0.