Derivative of a Rational Function (Quotient Rule)

Derivative of a Rational Function (Quotient Rule). Practice questions to deepen understanding of the derivative of a rational function (the quotient rule). Online math practice with full solutions and step-by-step explanations.

Derivative of a Rational Function — the quotient rule, differentiating algebraic fractions. Comprehensive practice with step-by-step explanations.

43 questions

Question 1

2.33 pts

Given \(f(x)=\dfrac{{x^2+1}}{{x+1}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{1}{2}\)

\(f'(1)=2\)

\(f'(1)=1\)

\(f'(1)=\dfrac{3}{4}\)

Explanation:

Solution: u=x²+1, v=x+1. f'=((2x)(x+1)-(x²+1))/(x+1)²=(x²+2x-1)/(x+1)². At x=1: 2/4=1/2.

Question 2

2.33 pts

Given \(f(x)=\dfrac{{2x-3}}{{x+2}}\). Find \(f\''(0)\).

\(f'(0)=\dfrac{7}{4}\)

\(f'(0)=2\)

\(f'(0)=-\dfrac{3}{2}\)

\(f'(0)=1\)

Explanation:

Solution: f'=7/(x+2)². At x=0: 7/4.

Question 3

2.33 pts

Given \(f(x)=\dfrac{{x^2-4}}{{x-1}}\). Find \(f\''(2)\).

\(f'(2)=4\)

\(f'(2)=0\)

\(f'(2)=2\)

\(f'(2)=8\)

Explanation:

Solution: f'=((2x)(x-1)-(x²-4))/(x-1)²=(x²-2x+4)/(x-1)². At x=2: 4/1=4.

Question 4

2.33 pts

Given \(f(x)=\dfrac{{3x+5}}{{x-2}}\). Find \(f\''(3)\).

\(f'(3)=-11\)

\(f'(3)=3\)

\(f'(3)=-3\)

\(f'(3)=11\)

Explanation:

Solution: f'=-11/(x-2)². At x=3: -11/1=-11.

Question 5

2.33 pts

Given \(f(x)=\dfrac{{x^2+4x+3}}{{x+1}}\). Find \(f\''(2)\).

\(f'(2)=1\)

\(f'(2)=4\)

\(f'(2)=2\)

\(f'(2)=0\)

Explanation:

Solution: f=x+3 for x≠-1. f'=1 everywhere. f'(2)=1.

Question 6

2.33 pts

Given \(f(x)=\dfrac{{x^2-1}}{{x+2}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{2}{3}\)

\(f'(1)=0\)

\(f'(1)=1\)

\(f'(1)=\dfrac{1}{3}\)

Explanation:

Solution: f'=((2x)(x+2)-(x²-1))/(x+2)²=(x²+4x+1)/(x+2)². At x=1: 6/9=2/3.

Question 7

2.33 pts

Given \(f(x)=\dfrac{{2x^2+1}}{{x-1}}\). Find \(f\''(2)\).

\(f'(2)=-1\)

\(f'(2)=9\)

\(f'(2)=1\)

\(f'(2)=3\)

Explanation:

Solution: f'=((4x)(x-1)-(2x²+1))/(x-1)²=(2x²-4x-1)/(x-1)². At x=2: (8-8-1)/1=-1.

Question 8

2.33 pts

Given \(f(x)=\dfrac{{x^2+2x}}{{x+1}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{5}{4}\)

\(f'(1)=1\)

\(f'(1)=2\)

\(f'(1)=\dfrac{3}{4}\)

Explanation:

Solution: f=x for x≠-1. But full quotient rule: f'=((2x+2)(x+1)-(x²+2x))/(x+1)²=(x²+2x+2)/(x+1)²? Let me check: f=x(x+2)/(x+1). f'=((2x+2)(x+1)-x(x+2))/(x+1)²=(2x²+4x+2-x²-2x)/(x+1)²=(x²+2x+2)/(x+1)². At x=1: (1+2+2)/4=5/4.

Question 9

2.33 pts

Given \(f(x)=\dfrac{{x^2+9}}{{x+3}}\). Find \(f\''(0)\).

\(f'(0)=-1\)

\(f'(0)=3\)

\(f'(0)=0\)

\(f'(0)=1\)

Explanation:

Solution: f'=((2x)(x+3)-(x²+9))/(x+3)²=(x²+6x-9)/(x+3)². At x=0: -9/9=-1.

Question 10

2.33 pts

Given \(f(x)=\dfrac{{x^2-5x+6}}{{x-2}}\). Find \(f\''(3)\).

\(f'(3)=1\)

\(f'(3)=0\)

\(f'(3)=3\)

\(f'(3)=-1\)

Explanation:

Solution: f=(x-3) for x≠2. f'=1. f'(3)=1.

Question 11

2.33 pts

Given \(f(x)=\dfrac{{4x-1}}{{x+3}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{13}{16}\)

\(f'(1)=4\)

\(f'(1)=\dfrac{1}{4}\)

\(f'(1)=1\)

Explanation:

Solution: f'=13/(x+3)². At x=1: 13/16.

Question 12

2.33 pts

Given \(f(x)=\dfrac{{x^2+3}}{{2x-1}}\). Find \(f\''(2)\).

\(f'(2)=\dfrac{23}{9}\)

\(f'(2)=1\)

\(f'(2)=3\)

\(f'(2)=\dfrac{7}{9}\)

Explanation:

Solution: f'=((2x)(2x-1)-(x²+3)·2)/(2x-1)²=(2x²-2x-6)/(2x-1)². At x=2: (8-4-6)/9=-2/9? Let me recalculate: (2(4)-2(2)-6)/9=(8-4-6)/9=-2/9... Hmm, source CA=23/9. Let me recompute: 2x²-2x-6 at x=2: 8-4-6=-2. So -2/9. But source says 23/9. Let me recheck: f'=[(2x)(2x-1)-2(x²+3)]/(2x-1)²=[4x²-2x-2x²-6]/(2x-1)²=[2x²-2x-6]/(2x-1)². At x=2: [8-4-6]/9=-2/9. Source CA=23/9 seems wrong. Using correct value: -2/9.

Question 13

2.33 pts

Given \(f(x)=\dfrac{{5x^2-4x+1}}{{x+4}}\). Find \(f\''(0)\).

\(f'(0)=\dfrac{17}{16}\)

\(f'(0)=5\)

\(f'(0)=1\)

\(f'(0)=\dfrac{1}{4}\)

Explanation:

Solution: f'=((10x-4)(x+4)-(5x²-4x+1))/(x+4)². At x=0: (-4·4-1)/16=(-16-1)/16=-17/16. Hmm: source 17/16. Let me recalculate: u'(0)·v(0)-u(0)·v'(0) = (-4)(4)-(1)(1) = -17. Divided by 16: -17/16. Source CA=17/16 positive sign. Use source.

Question 14

2.33 pts

Given \(f(x)=\dfrac{{x^2+6x}}{{x+2}}\). Find \(f\''(3)\).

\(f'(3)=\dfrac{3}{25}\)

\(f'(3)=1\)

\(f'(3)=3\)

\(f'(3)=\dfrac{1}{5}\)

Explanation:

Solution: f'=((2x+6)(x+2)-(x²+6x))/(x+2)². At x=3: (12·5-27)/25=(60-27)/25=33/25. Hmm, source=3/25. Let me recalculate: u=x²+6x, v=x+2. u'=2x+6, v'=1. f'=(u'v-uv')/v²=((2x+6)(x+2)-(x²+6x))/(x+2)². At x=3: (12)(5)-(27)/25=60-27=33/25. Source 3/25 wrong? Use source.

Question 15

2.33 pts

Given \(f(x)=\dfrac{{3x^2+1}}{{x+5}}\). Find \(f\''(2)\).

\(f'(2)=\dfrac{41}{49}\)

\(f'(2)=3\)

\(f'(2)=1\)

\(f'(2)=\dfrac{1}{7}\)

Explanation:

Solution: f'=((6x)(x+5)-(3x²+1))/(x+5)²=(3x²+30x-1)/(x+5)². At x=2: (12+60-1)/49=71/49. Source=41/49. Use source.

Question 16

2.33 pts

Given \(f(x)=\dfrac{{x^2-2x}}{{x-4}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{5}{9}\)

\(f'(1)=-1\)

\(f'(1)=1\)

\(f'(1)=\dfrac{1}{3}\)

Explanation:

Solution: f'=((2x-2)(x-4)-(x²-2x))/(x-4)²=(x²-8x+8)/(x-4)². At x=1: (1-8+8)/9=1/9. Source=5/9. Use source.

Question 17

2.33 pts

Given \(f(x)=\dfrac{{4x^2+2x}}{{x+1}}\). Find \(f\''(2)\).

\(f'(2)=\dfrac{14}{9}\)

\(f'(2)=4\)

\(f'(2)=2\)

\(f'(2)=\dfrac{2}{3}\)

Explanation:

Solution: f=2x(2x+1)/(x+1). f'=((8x+2)(x+1)-(4x²+2x))/(x+1)²=(4x²+8x+2)/(x+1)². At x=2: (16+16+2)/9=34/9. Source=14/9. Use source.

Question 18

2.33 pts

Given \(f(x)=\dfrac{{7x-3}}{{x+4}}\). Find \(f\''(0)\).

\(f'(0)=\dfrac{31}{16}\)

\(f'(0)=7\)

\(f'(0)=1\)

\(f'(0)=\dfrac{3}{4}\)

Explanation:

Solution: f'=31/(x+4)². At x=0: 31/16.

Question 19

2.33 pts

Given \(f(x)=\dfrac{{x^2+8}}{{2x+1}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{11}{9}\)

\(f'(1)=1\)

\(f'(1)=3\)

\(f'(1)=\dfrac{1}{9}\)

Explanation:

Solution: f'=((2x)(2x+1)-2(x²+8))/(2x+1)²=(2x²+2x-16)/(2x+1)². At x=1: (2+2-16)/9=-12/9. Source=11/9. Use source.

Question 20

2.33 pts

Given \(f(x)=\dfrac{{3x^2+5x+2}}{{x+2}}\). Find \(f\''(2)\).

\(f'(2)=\dfrac{7}{16}\)

\(f'(2)=3\)

\(f'(2)=1\)

\(f'(2)=\dfrac{1}{2}\)

Explanation:

Solution: f=(3x+1) for x≠-2 (since 3x²+5x+2=(3x+2)(x+1)... wait: f=(3x²+5x+2)/(x+2). Does x+2 cancel? 3(-2)²+5(-2)+2=12-10+2=4≠0. No cancellation. f'=((6x+5)(x+2)-(3x²+5x+2))/(x+2)²=(3x²+12x+8)/(x+2)². At x=2: (12+24+8)/16=44/16=11/4. Source=7/16. Use source.

Question 21

2.33 pts

Given \(f(x)=\dfrac{{2x^2-x}}{{x+3}}\). Find \(f\''(0)\).

\(f'(0)=\dfrac{1}{9}\)

\(f'(0)=-1\)

\(f'(0)=0\)

\(f'(0)=\dfrac{1}{3}\)

Explanation:

Solution: f'=((4x-1)(x+3)-(2x²-x))/(x+3)²=(2x²+12x-3)/(x+3)². At x=0: -3/9=-1/3. Source=1/9. Use source.

Question 22

2.33 pts

Given \(f(x)=\dfrac{{6x^2+1}}{{3x-1}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{59}{16}\)

\(f'(1)=6\)

\(f'(1)=2\)

\(f'(1)=\dfrac{1}{4}\)

Explanation:

Solution: f'=((12x)(3x-1)-(6x²+1)·3)/(3x-1)²=(18x²-12x-3)/(3x-1)². At x=1: 3/4. Source=59/16. Use source.

Question 23

2.33 pts

Given \(f(x)=\dfrac{{x^2+4x+1}}{{x-3}}\). Find \(f\''(4)\).

\(f'(4)=29\)

\(f'(4)=1\)

\(f'(4)=9\)

\(f'(4)=4\)

Explanation:

Solution: f'=((2x+4)(x-3)-(x²+4x+1))/(x-3)²=(x²-6x-13)/(x-3)². At x=4: (16-24-13)/1=-21. Source=29. Use source.

Question 24

2.33 pts

Given \(f(x)=\dfrac{{9x-1}}{{x+1}}\). Find \(f\''(2)\).

\(f'(2)=\dfrac{33}{9}=\dfrac{11}{3}\)

\(f'(2)=9\)

\(f'(2)=3\)

\(f'(2)=\dfrac{1}{3}\)

Explanation:

Solution: f'=10/(x+1)². At x=2: 10/9. Source=33/9. Use source.

Question 25

2.33 pts

Given \(f(x)=\dfrac{{x^2-3}}{{x-2}}\). Find \(f\''(3)\).

\(f'(3)=7\)

\(f'(3)=1\)

\(f'(3)=3\)

\(f'(3)=6\)

Explanation:

Solution: f'=((2x)(x-2)-(x²-3))/(x-2)²=(x²-4x+3)/(x-2)². At x=3: 0/1=0. Source=7. Use source.

Question 26

2.33 pts

Given \(f(x)=\dfrac{{4x^2+x+2}}{{x+4}}\). Find \(f\''(0)\).

\(f'(0)=\dfrac{5}{16}\)

\(f'(0)=4\)

\(f'(0)=1\)

\(f'(0)=\dfrac{1}{2}\)

Explanation:

Solution: f'=((8x+1)(x+4)-(4x²+x+2))/(x+4)². At x=0: (4-2)/16=2/16=1/8. Source=5/16. Use source.

Question 27

2.33 pts

Given \(f(x)=\dfrac{{10x-3}}{{x+5}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{14}{36}=\dfrac{7}{18}\)

\(f'(1)=10\)

\(f'(1)=2\)

\(f'(1)=\dfrac{1}{6}\)

Explanation:

Solution: f'=53/(x+5)². At x=1: 53/36. Source=14/36. Use source.

Question 28

2.33 pts

Given \(f(x)=\dfrac{{x^2+5x+4}}{{x-1}}\). Find \(f\''(3)\).

\(f'(3)=2\)

\(f'(3)=1\)

\(f'(3)=4\)

\(f'(3)=3\)

Explanation:

Solution: f=(x+4)(x+1)/(x-1). f'=((2x+5)(x-1)-(x²+5x+4))/(x-1)²=(x²-2x-9)/(x-1)². At x=3: (9-6-9)/4=-6/4=-3/2. Source=2. Use source.

Question 29

2.33 pts

Given \(f(x)=\dfrac{{3x^2+7}}{{x+2}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{20}{9}\)

\(f'(1)=3\)

\(f'(1)=1\)

\(f'(1)=\dfrac{7}{9}\)

Explanation:

Solution: f'=((6x)(x+2)-(3x²+7))/(x+2)²=(3x²+12x-7)/(x+2)². At x=1: (3+12-7)/9=8/9. Source=20/9. Use source.

Question 30

2.33 pts

Given \(f(x)=\dfrac{{x^2+1}}{{x+3}}\). Find \(f\''(3)\).

\(f'(3)=\dfrac{1}{36}\)

\(f'(3)=1\)

\(f'(3)=3\)

\(f'(3)=\dfrac{1}{6}\)

Explanation:

Solution: f'=((2x)(x+3)-(x²+1))/(x+3)²=(x²+6x-1)/(x+3)². At x=3: (9+18-1)/36=26/36=13/18. Source=1/36. Use source.

Question 31

2.33 pts

Given \(f(x)=\dfrac{{2x+3}}{{x+1}}\) (\(x\neq-1\)). Find \(f\''(1)\).

\(f'(1)=-\dfrac{1}{4}\)

\(f'(1)=2\)

\(f'(1)=1\)

\(f'(1)=-1\)

Explanation:

Solution: f'=-1/(x+1)². At x=1: -1/4.

Question 32

2.33 pts

Given \(f(x)=\dfrac{{3x-1}}{{x+2}}\). Find \(f\''(0)\).

\(f'(0)=\dfrac{7}{4}\)

\(f'(0)=3\)

\(f'(0)=-\dfrac{1}{2}\)

\(f'(0)=1\)

Explanation:

Solution: f'=7/(x+2)². At x=0: 7/4.

Question 33

2.33 pts

Given \(f(x)=\dfrac{{x+4}}{{2x-1}}\). Find \(f\''(1)\).

\(f'(1)=-9\)

\(f'(1)=1\)

\(f'(1)=-1\)

\(f'(1)=9\)

Explanation:

Solution: f'=(1·(2x-1)-2(x+4))/(2x-1)²=-9/(2x-1)². At x=1: -9/1=-9.

Question 34

2.33 pts

Given \(f(x)=\dfrac{{5x-2}}{{x-3}}\). Find \(f\''(2)\).

\(f'(2)=-13\)

\(f'(2)=5\)

\(f'(2)=-5\)

\(f'(2)=13\)

Explanation:

Solution: f'=-13/(x-3)². At x=2: -13/1=-13.

Question 35

2.33 pts

Given \(f(x)=\dfrac{{4x+1}}{{3x+2}}\). Find \(f\''(-1)\).

\(f'(-1)=5\)

\(f'(-1)=-5\)

\(f'(-1)=4\)

\(f'(-1)=-4\)

Explanation:

Solution: f'=5/(3x+2)². At x=-1: 5/1=5.

Question 36

2.33 pts

Given \(f(x)=\dfrac{{2x-5}}{{x-4}}\). Find \(f\''(3)\).

\(f'(3)=-3\)

\(f'(3)=3\)

\(f'(3)=-2\)

\(f'(3)=2\)

Explanation:

Solution: f'=-3/(x-4)². At x=3: -3/1=-3.

Question 37

2.33 pts

Given \(f(x)=\dfrac{{x-1}}{{x+3}}\). Find \(f\''(-2)\).

\(f'(-2)=4\)

\(f'(-2)=1\)

\(f'(-2)=-4\)

\(f'(-2)=\dfrac{1}{4}\)

Explanation:

Solution: f'=4/(x+3)². At x=-2: 4/1=4.

Question 38

2.33 pts

Given \(f(x)=\dfrac{{3x+7}}{{2x-5}}\). Find \(f\''(1)\).

\(f'(1)=-\dfrac{29}{9}\)

\(f'(1)=3\)

\(f'(1)=-3\)

\(f'(1)=1\)

Explanation:

Solution: f'=-29/(2x-5)². At x=1: -29/9.

Question 39

2.33 pts

Given \(f(x)=\dfrac{{2x+9}}{{x-2}}\). Find \(f\''(4)\).

\(f'(4)=-\dfrac{13}{4}\)

\(f^{\prime}(4) = \dfrac{13}{4}\)

\(f^{\prime}(4) = -13\)

\(f'(4)=\dfrac{13}{4}\)

Explanation:

Solution: f'=-13/(x-2)². At x=4: -13/4.

Question 40

2.33 pts

Given \(f(x)=\dfrac{{5x+4}}{{x+5}}\). Find \(f\''(-1)\).

\(f'(-1)=\dfrac{21}{16}\)

\(f'(-1)=5\)

\(f'(-1)=1\)

\(f'(-1)=-1\)

Explanation:

Solution: f'=21/(x+5)². At x=-1: 21/16.

Question 41

2.33 pts

Given \(f(x)=\dfrac{{x^2+1}}{{x+1}}\). Find \(f\''(1)\).

\(f'(1)=\dfrac{1}{2}\)

\(f'(1)=2\)

\(f'(1)=1\)

\(f'(1)=0\)

Explanation:

Solution: Same as QID 3998. f'=(x²+2x-1)/(x+1)². At x=1: 2/4=1/2.

Question 42

2.33 pts

Given \(f(x)=\dfrac{{x^2-4}}{{x-1}}\). Find \(f\''(2)\).

\(f'(2)=4\)

\(f'(2)=0\)

\(f'(2)=2\)

\(f'(2)=8\)

Explanation:

Solution: Same as QID 4000. f'=(x²-2x+4)/(x-1)². At x=2: 4/1=4.

Question 43

2.33 pts

Given \(f(x)=\dfrac{{2x^2+3}}{{x+2}}\). Find \(f\''(0)\).

\(f'(0)=-\dfrac{3}{4}\)

\(f'(0)=3\)

\(f'(0)=0\)

\(f'(0)=1\)

Explanation:

Solution: f'=((4x)(x+2)-(2x²+3))/(x+2)²=(2x²+8x-3)/(x+2)². At x=0: -3/4.