Differentiation

Maths, HSSC-Punjab First Year. This chapter is split into 35 topics and has 1,029 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Equation of the tangent line at a point 30 questions
2. Structural stress and engineering rates 30 questions
3. Power rule with fractional exponents and roots 30 questions
4. Derivative of a sum or difference of functions 30 questions
5. Exercise 13.2 differentiation problems 30 questions
6. Difference quotient and average rate of change 30 questions
7. Derivative of reciprocal powers by definition 30 questions
8. Differentiability and continuity 30 questions
9. The four steps of differentiation by definition 30 questions
10. Multiplying out before differentiating 30 questions
11. Particle at rest and acceleration zero 30 questions
12. Rates of change in physical models 30 questions
13. Quotient rule statement and proof 30 questions
14. Average velocity over a time interval 30 questions
15. Notations for the derivative 30 questions
16. Quotient rule at a point 30 questions
17. Instantaneous velocity as a derivative 30 questions
18. Acceleration as the derivative of velocity second derivative 30 questions
19. Rate of change of a financial investment 30 questions
20. Product rule at a point 29 questions
21. Derivative of the square root by definition 24 questions
22. Derivative of a constant multiple of a function 30 questions
23. Differentiability of piecewise and absolute value functions 30 questions
24. Marginal profit cost and revenue 30 questions
25. Value of the derivative of a polynomial at a point 30 questions
26. Velocity and acceleration of a particle 30 questions
27. Product rule statement and proof 30 questions
28. Tangent and secant lines to a curve 23 questions
29. Gradient of the tangent line by the limit definition 30 questions
30. Derivation of the power rule for positive integers 30 questions
31. Review of gradients and tangents of polynomial graphs 30 questions
32. Horizontal tangents and prescribed gradients 30 questions
33. Differentiating polynomial functions 30 questions
34. Power rule for negative and rational exponents 30 questions
35. Derivatives of simple functions by definition 23 questions

Sample questions: Differentiation

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. In the seventeenth century differential calculus was invented independently by

• A. Archimedes and Euclid, in ancient Greece
• B. Isaac Newton and Leonhard Euler, working together as a team
• C. Isaac Newton in England and Gottfried Leibniz in Germany(correct answer)
• D. Pierre de Fermat and Rene Descartes, in the sixteenth century

Answer: C.The introduction of Unit 13 credits the English mathematician Newton (1642-1727) and the German mathematician Leibniz (1646-1716) with the independent invention of differential calculus.

Q2. Using the quotient rule, differentiate y = (2x³ − 3x² − x)/(x − 3) with respect to x.

• A. (6x² − 6x − 1)/(x − 3)²
• B. (4x³ − 21x² + 18x + 3)/(x − 3)²(correct answer)
• C. (4x³ − 21x² + 18x + 6)/(x − 3)²
• D. (4x³ − 21x² + 18x + 3)/(2x³ − 3x² − x)²

Answer: B.d/dx (f/g) = (f′g − fg′)/g² with f = 2x³ − 3x² − x, g = x − 3: numerator = (6x² − 6x − 1)(x − 3) − (2x³ − 3x² − x)(1) = 4x³ − 21x² + 18x + 3, so dy/dx = (4x³ − 21x² + 18x + 3)/(x − 3)².

Q3. Using the quotient rule, differentiate y = (3x² + 4x + 2)/(x² − 3x + 1) with respect to x.

• A. (−13x² + 2x + 10)/(x² − 3x + 1)²(correct answer)
• B. (−13x² + 2x + 10)/(x² − 3x + 1)
• C. (−13x² + 2x + 10)/(3x² + 4x + 2)²
• D. (13x² − 2x − 10)/(x² − 3x + 1)²

Answer: A.d/dx (f/g) = (f′g − fg′)/g² with f = 3x² + 4x + 2, g = x² − 3x + 1: numerator = (6x + 4)(x² − 3x + 1) − (3x² + 4x + 2)(2x − 3) = −13x² + 2x + 10, so dy/dx = (−13x² + 2x + 10)/(x² − 3x + 1)².

Q4. Using the quotient rule, differentiate y = (3x² − 4x + 4)/(−x − 2) with respect to x.

• A. (−9x² − 4x + 4)/(−x − 2)²
• B. (−3x² − 12x + 12)/(−x − 2)²(correct answer)
• C. (−3x² − 12x + 15)/(−x − 2)²
• D. (−3x² − 12x + 12)/(3x² − 4x + 4)²

Answer: B.d/dx (f/g) = (f′g − fg′)/g² with f = 3x² − 4x + 4, g = −x − 2: numerator = (6x − 4)(−x − 2) − (3x² − 4x + 4)(−1) = −3x² − 12x + 12, so dy/dx = (−3x² − 12x + 12)/(−x − 2)².

Q5. Using the quotient rule, differentiate y = (2x² + 3x − 2)/(2x² + 2x − 2) with respect to x.

• A. (−2x² − 2)/(2x² + 2x − 2)
• B. (16x³ + 30x² − 4x − 10)/(2x² + 2x − 2)²
• C. (4x + 3)/(4x + 2)
• D. (−2x² − 2)/(2x² + 2x − 2)²(correct answer)

Answer: D.d/dx (f/g) = (f′g − fg′)/g² with f = 2x² + 3x − 2, g = 2x² + 2x − 2: numerator = (4x + 3)(2x² + 2x − 2) − (2x² + 3x − 2)(4x + 2) = −2x² − 2, so dy/dx = (−2x² − 2)/(2x² + 2x − 2)².

Q6. Using the quotient rule, differentiate y = (−x³ − 2x² + 4)/(−2x² + 2x − 3) with respect to x.

• A. (−2x⁴ + 4x³ − 5x² − 28x + 8)/(−2x² + 2x − 3)²
• B. (2x⁴ − 4x³ + 5x² + 28x − 8)/(−x³ − 2x² + 4)²
• C. (2x⁴ − 4x³ + 5x² + 28x − 6)/(−2x² + 2x − 3)²
• D. (2x⁴ − 4x³ + 5x² + 28x − 8)/(−2x² + 2x − 3)²(correct answer)

Answer: D.d/dx (f/g) = (f′g − fg′)/g² with f = −x³ − 2x² + 4, g = −2x² + 2x − 3: numerator = (−3x² − 4x)(−2x² + 2x − 3) − (−x³ − 2x² + 4)(2 − 4x) = 2x⁴ − 4x³ + 5x² + 28x − 8, so dy/dx = (2x⁴ − 4x³ + 5x² + 28x − 8)/(−2x² + 2x − 3)².

Q7. Using the quotient rule, differentiate y = (−2x² − 2x + 4)/(2x² + x + 3) with respect to x.

• A. (2x² − 28x − 10)/(2x² + x + 3)²(correct answer)
• B. (2x² − 28x − 10)/(2x² + x + 3)
• C. (−16x² − 12x − 2)/(2x² + x + 3)²
• D. (−2x² + 28x + 10)/(2x² + x + 3)²

Answer: A.d/dx (f/g) = (f′g − fg′)/g² with f = −2x² − 2x + 4, g = 2x² + x + 3: numerator = (−4x − 2)(2x² + x + 3) − (−2x² − 2x + 4)(4x + 1) = 2x² − 28x − 10, so dy/dx = (2x² − 28x − 10)/(2x² + x + 3)².

Q8. Using the quotient rule, differentiate y = (−3x² + 2x − 2)/(2x + 1) with respect to x.

• A. (−6x² − 6x + 6)/(2x + 1)
• B. (−18x² + 2x − 2)/(2x + 1)²
• C. (6x² + 6x − 6)/(2x + 1)²
• D. (−6x² − 6x + 6)/(2x + 1)²(correct answer)

Answer: D.d/dx (f/g) = (f′g − fg′)/g² with f = −3x² + 2x − 2, g = 2x + 1: numerator = (2 − 6x)(2x + 1) − (−3x² + 2x − 2)(2) = −6x² − 6x + 6, so dy/dx = (−6x² − 6x + 6)/(2x + 1)².

Q9. Using the quotient rule, differentiate y = (−2x³ + x² − 4x)/(x² + 3) with respect to x.

• A. (−2x⁴ − 14x² + 6x − 12)/(−2x³ + x² − 4x)²
• B. (−2x⁴ − 14x² + 6x − 12)/(x² + 3)²(correct answer)
• C. 1/(2x)(−6x² + 2x − 4)
• D. (2x⁴ + 14x² − 6x + 12)/(x² + 3)²

Answer: B.d/dx (f/g) = (f′g − fg′)/g² with f = −2x³ + x² − 4x, g = x² + 3: numerator = (−6x² + 2x − 4)(x² + 3) − (−2x³ + x² − 4x)(2x) = −2x⁴ − 14x² + 6x − 12, so dy/dx = (−2x⁴ − 14x² + 6x − 12)/(x² + 3)².

Q10. Using the quotient rule, differentiate y = (x² − 3x + 3)/(2x² + 1) with respect to x.

• A. (6x² − 10x − 4)/(2x² + 1)²
• B. (8x² − 12x)/(2x² + 1)²
• C. (6x² − 10x − 3)/(2x² + 1)²(correct answer)
• D. (8x³ − 18x² + 14x − 3)/(2x² + 1)²

Answer: C.d/dx (f/g) = (f′g − fg′)/g² with f = x² − 3x + 3, g = 2x² + 1: numerator = (2x − 3)(2x² + 1) − (x² − 3x + 3)(4x) = 6x² − 10x − 3, so dy/dx = (6x² − 10x − 3)/(2x² + 1)².