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Worksheets

Differentiation and Integration Topics

Total questions: 100

Worksheet time: 50mins

Name
Class
Date
1.

What is the topic covered in section 4.1?

a)

DIFFERENTIATION

b)

INTEGRATION

c)

TRIGONOMETRY

d)

PROBABILITY

2.

What is the topic covered in section 4.2?

a)

DIFFERENTIATION OF BASIC FUNCTIONS

b)

INTEGRATION OF BASIC FUNCTIONS

c)

APPLICATIONS OF DERIVATIVES

d)

LIMITS AND CONTINUITY

3.

What type of functions are covered in section 4.2.1?

a)

POLYNOMIAL FUNCTIONS

b)

EXPONENTIAL FUNCTIONS

c)

LOGARITHMIC FUNCTIONS

d)

TRIGONOMETRIC FUNCTIONS

4.

What type of functions are covered in section 4.2.2?

a)

EXPONENTIAL FUNCTIONS

b)

LINEAR FUNCTIONS

c)

QUADRATIC FUNCTIONS

d)

LOGARITHMIC FUNCTIONS

5.

What type of functions are covered in section 4.2.3?

a)

LOGARITHMIC FUNCTIONS

b)

TRIGONOMETRIC FUNCTIONS

c)

POLYNOMIAL FUNCTIONS

d)

EXPONENTIAL FUNCTIONS

6.

What type of functions are covered in section 4.2.4?

a)

TRIGONOMETRY FUNCTIONS

b)

LOGARITHMIC FUNCTIONS

c)

EXPONENTIAL FUNCTIONS

d)

POLYNOMIAL FUNCTIONS

7.

What rule is discussed in section 4.2.5?

a)

SUM AND DIFFERENCE RULE OF BASIC FUNCTIONS

b)

PRODUCT RULE OF BASIC FUNCTIONS

c)

CHAIN RULE OF BASIC FUNCTIONS

d)

QUOTIENT RULE OF BASIC FUNCTIONS

8.

What is the topic covered in section 4.3?

a)

TECHNIQUE OF DIFFERENTIATIONS

b)

INTEGRATION METHODS

c)

APPLICATIONS OF DERIVATIVES

d)

LIMITS AND CONTINUITY

9.

What rule is discussed in section 4.3.1?

a)

PRODUCT RULE

b)

SUM RULE

c)

CHAIN RULE

d)

QUOTIENT RULE

10.

What rule is discussed in section 4.3.2?

a)

QUOTIENT RULE

b)

PRODUCT RULE

c)

CHAIN RULE

d)

POWER RULE

11.

What rule is discussed in section 4.3.3?

a)

CHAIN RULE

b)

PRODUCT RULE

c)

QUOTIENT RULE

d)

POWER RULE

12.

What is the topic covered in section 4.4?

a)

INTEGRATION

b)

DIFFERENTIATION

c)

PROBABILITY

d)

GEOMETRY

13.

What is the topic covered in section 4.5?

a)

TYPE OF INTEGRATION

b)

APPLICATION OF INTEGRATION

c)

DEFINITION OF INTEGRATION

d)

METHODS OF DIFFERENTIATION

14.

What type of integral is covered in section 4.5.1?

a)

INDEFINITE INTEGRAL

b)

DEFINITE INTEGRAL

c)

IMPROPER INTEGRAL

d)

LINE INTEGRAL

15.

What type of integral is covered in section 4.5.2?

a)

DEFINITE INTEGRAL

b)

INDEFINITE INTEGRAL

c)

IMPROPER INTEGRAL

d)

LINE INTEGRAL

16.

What is the topic covered in section 4.6?

a)

INTEGRATION OF BASIC FUNCTIONS

b)

DIFFERENTIATION OF BASIC FUNCTIONS

c)

APPLICATIONS OF DERIVATIVES

d)

LIMITS AND CONTINUITY

17.

What method is discussed in section 4.7.1?

a)

SUBSTITUTION METHOD

b)

DIVISION METHOD

c)

ITERATION METHOD

d)

RECURSION METHOD

18.

What method is discussed in section 4.7.2?

a)

TABULAR METHOD

b)

GRAPHICAL METHOD

c)

NUMERICAL METHOD

d)

ANALYTICAL METHOD

19.

What method is discussed in section 4.7.3?

a)

PARTIAL FRACTIONS

b)

Integration by Substitution

c)

Trigonometric Integrals

d)

Numerical Integration

20.

What is the derivative of y = f(x)?

a)

y' or f'

b)

y'' or f''

c)

y(x) or f(x)

d)

y + f

21.

Fill in the blank: The derivative of y = xnx^n is _________.

a)

nxn−1nx^{n-1}

b)

nxn^x

c)

xnx^{n}

d)

n∗xnn*x^{n}

22.

Fill in the blank: If y = cu, then dydx=\frac{dy}{dx} = _________.

a)

c dudx\frac{du}{dx}

b)

dcdx⋅u\frac{dc}{dx} \cdot u

c)

c dudc\frac{du}{dc}

d)

u dcdx\frac{dc}{dx}

23.

If y = c, where c is a constant, then dydx=0\frac{dy}{dx} = 0 .

a)

True

b)

False

24.

Find the derivative of the following function: a) y = x15x^{15} Write your answer as dy/dx = ____

a)

15x1415x^{14}

b)

14x1514x^{15}

c)

x14x^{14}

d)

15x1515x^{15}

25.

Find the derivative of the following function: b) y = −3x−4-3x^{-4} Write your answer as dy/dx = ____

a)

12x−512x^{-5}

b)

−12x−5-12x^{-5}

c)

−3x−3-3x^{-3}

d)

3x−43x^{-4}

26.

Find the derivative of the following function: c) y = 133\frac{1}{\sqrt[3]{3}} Write your answer as dy/dx = ____

a)

0

b)

1

c)

13\frac{1}{3}

d)

-1

27.

Find the derivative of the following function: d) y = 5x1/35x^{1/3} Write your answer as dy/dx = ____

a)

53(x23)\frac{5}{3(\sqrt[3]{x^2})}

b)

53(x3)\frac{5}{3(\sqrt[3]{x})}

c)

5x2/35x^{2/3}

d)

53x2/3\frac{5}{3x^{2/3}}

28.

Find the derivative of the following function: y = 3x53x^{5} Write your answer as dy/dx = ____

a)

15x415x^{4}

b)

5x45x^{4}

c)

3x43x^{4}

d)

18x518x^{5}

29.

Find the derivative of the following function: f) y = x−25\frac{x^{-2}}{5} Write your answer as dy/dx = ____

a)

−25x3-\frac{2}{5x^{3}}

b)

25x3\frac{2}{5x^{3}}

c)

−25x2-\frac{2}{5x^{2}}

d)

25x2\frac{2}{5x^{2}}

30.

Find the derivative of the following function: y = e9xe^{9x}

a)

9e9x9e^{9x}

b)

e9xe^{9x}

c)

exe^{x}

d)

9ex9e^{x}

31.

Find the derivative of the following function: b) y = −4e7x-4e^{7x} Write your answer as dy/dx = ____

a)

−28e7x-28e^{7x}

b)

−4e7x-4e^{7x}

c)

28e7x28e^{7x}

d)

−28ex-28e^{x}

32.

Find the derivative of the following function: c) y = −3e−3x-3e^{-3x} Write your answer as dy/dx = ____

a)

9e−3x9e^{-3x}

b)

−3e−3x-3e^{-3x}

c)

−9e−3x-9e^{-3x}

d)

3e−3x3e^{-3x}

33.

Find the derivative of the following function: y = −23e6x-\frac{2}{3}e^{6x} Write your answer as dy/dx = ____

a)

−4e6x-4e^{6x}

b)

−2e6x-2e^{6x}

c)

−123e6x-\frac{12}{3}e^{6x}

d)

−23ex-\frac{2}{3}e^{x}

34.

Find the derivative of the following function: y = 35e−13x\frac{3}{5}e^{-\frac{1}{3}x} Write your answer as dy/dx = ____

a)

−15e−13x-\frac{1}{5}e^{-\frac{1}{3}x}

b)

15e−13x\frac{1}{5}e^{-\frac{1}{3}x}

c)

−35e−13x-\frac{3}{5}e^{-\frac{1}{3}x}

d)

−13e−13x-\frac{1}{3}e^{-\frac{1}{3}x}

35.

Find the derivative of the following function: f) y = Write your answer as dy/dx = ____

a)

−83e−4x-\frac{8}{3}e^{-4x}

b)

83e−4x\frac{8}{3}e^{-4x}

c)

−23e−4x-\frac{2}{3}e^{-4x}

d)

23e4x\frac{2}{3}e^{4x}

36.

Find the derivative of the following function: g) y = 9x9^{x}

a)

9xln⁡99^{x}\ln 9

b)

9xln⁡x9^{x}\ln x

c)

x⋅9xx\cdot 9^{x}

d)

9x

37.

Find the derivative of the following function: y = 23x2^{3x} Write your answer as dy/dx = ____

a)

23xln⁡2⋅32^{3x}\ln 2 \cdot 3

b)

23xln⁡3⋅22^{3x}\ln 3 \cdot 2

c)

32xln⁡2⋅33^{2x}\ln 2 \cdot 3

d)

23xln⁡62^{3x}\ln 6

38.

Find the derivative: dydx=−15e−13x\frac{dy}{dx} = -\frac{1}{5} e^{-\frac{1}{3}x}

a)

−15e−13x-\frac{1}{5} e^{-\frac{1}{3}x}

b)

c)

−13e−15x-\frac{1}{3} e^{-\frac{1}{5}x}

d)

−15e13x-\frac{1}{5} e^{\frac{1}{3}x}

39.

Find the derivative: dydx=−83e−4x/3\frac{dy}{dx} = -\frac{8}{3} e^{-4x/3}

a)

−83e−4x/3-\frac{8}{3} e^{-4x/3}

b)

83e−4x/3\frac{8}{3} e^{-4x/3}

c)

−83e4x/3-\frac{8}{3} e^{4x/3}

d)

−8e−4x/3-8 e^{-4x/3}

40.

Find the derivative: dydx=9xln⁡(9)\frac{dy}{dx} = 9^x \ln(9)

a)

9xln⁡(9)9^x \ln(9)

b)

9x9^x

c)

x 9^x ln⁡(9)\ln(9)

d)

9x \ln(9)

41.

Find the derivative: dydx=23xln⁡(2)(3)\frac{dy}{dx} = 2^{3x} \ln(2)(3)

a)

23xln⁡(2)(3)2^{3x} \ln(2)(3)

b)

23xln⁡(3)(2)2^{3x} \ln(3)(2)

c)

32xln⁡(2)(3)3^{2x} \ln(2)(3)

d)

23xln⁡(2)2^{3x} \ln(2)

42.

Find the derivative of the following function: y=ln⁡(2x)y = \ln(2x)

a)

dydx=1x\frac{dy}{dx} = \frac{1}{x}

b)

dydx=12x\frac{dy}{dx} = \frac{1}{2x}

c)

dydx=2x\frac{dy}{dx} = 2x

d)

dydx=ln⁡(2x)\frac{dy}{dx} = \ln(2x)

43.

Find the derivative of the following function: y=−3ln⁡(4x−1)y = -3 \ln(4x-1)

a)

dydx=−124x−1\frac{dy}{dx} = -\frac{12}{4x-1}

b)

dydx=−34x−1\frac{dy}{dx} = -\frac{3}{4x-1}

c)

dydx=−34x−1⋅4\frac{dy}{dx} = -\frac{3}{4x-1} \cdot 4

d)

dydx=−34x−1+4\frac{dy}{dx} = -\frac{3}{4x-1} + 4

44.

Find the derivative of the following function: y=−4ln⁡(3x)y = -4 \ln(3x)

a)

dydx=−4x\frac{dy}{dx} = -\frac{4}{x}

b)

dydx=−43x\frac{dy}{dx} = -\frac{4}{3x}

c)

dydx=−4x2\frac{dy}{dx} = -\frac{4}{x^2}

d)

dydx=−43x2\frac{dy}{dx} = -\frac{4}{3x^2}

45.

Find the derivative of the following function: y=−57ln⁡(7x)y = -\frac{5}{7} \ln(7x)

a)

dydx=−57x\frac{dy}{dx} = -\frac{5}{7x}

b)

dydx=−57x\frac{dy}{dx} = -\frac{5}{7}x

c)

dydx=−57ln⁡(x)\frac{dy}{dx} = -\frac{5}{7}\ln(x)

d)

dydx=−57x2\frac{dy}{dx} = -\frac{5}{7x^2}

46.

Find the derivative of the following function: y=15ln⁡(x2−1)y = \frac{1}{5} \ln(x^2-1)

a)

dydx=2x5(x2−1)\frac{dy}{dx} = \frac{2x}{5(x^2-1)}

b)

dydx=15(x2−1)\frac{dy}{dx} = \frac{1}{5(x^2-1)}

c)

dydx=25(x2−1)\frac{dy}{dx} = \frac{2}{5(x^2-1)}

d)

dydx=x5(x2−1)\frac{dy}{dx} = \frac{x}{5(x^2-1)}

47.

Find the derivative of the following function: y=−12ln⁡(4x2+3)y = -\frac{1}{2} \ln(4x^2+3)

a)

dydx=−4x4x2+3\frac{dy}{dx} = -\frac{4x}{4x^2+3}

b)

dydx=−2x4x2+3\frac{dy}{dx} = -\frac{2x}{4x^2+3}

c)

dydx=−14x2+3\frac{dy}{dx} = -\frac{1}{4x^2+3}

d)

dydx=−8x4x2+3\frac{dy}{dx} = -\frac{8x}{4x^2+3}

48.

Find the derivative of the following function: a) y = sin(2x) Fill in the blank: dy/dx = _______

a)

2 cos(2x)

b)

cos(2x)

c)

2 sin(2x)

d)

sin(2x)

49.

Find the derivative of the following function: b) y = -1/4 cos(3x) Fill in the blank: dy/dx = _______

a)

3/4 sin(3x)

b)

-3/4 sin(3x)

c)

-1/4 sin(3x)

d)

3/4 cos(3x)

50.

Find the derivative of the following function: c) y = 3/5 sin(10x) Fill in the blank: dy/dx = _______

a)

6 cos(10x)

b)

30 cos(10x)

c)

3/5 cos(10x)

d)

3/5 sin(10x)

51.

Find the derivative of the following function: d) y = -5 cos(4x) Fill in the blank: dy/dx = _______

a)

20 sin(4x)

b)

-20 sin(4x)

c)

-5 sin(4x)

d)

5 cos(4x)

52.

Find the derivative of the following function: y = 12sin(2x2+3)\frac{1}{2} sin(2x^2 + 3) Fill in the blank: dy/dx = _______

a)

2x cos(2x^2 + 3)

b)

cos(2x2+3)cos(2x^2 + 3)

c)

4x cos(2x^2 + 3)

d)

2x sin(2x^2 + 3)

53.

Find the derivative of the following function: f) y = 7 cos(3 - 5x) Fill in the blank: dy/dx = _______

a)

35 sin(3 - 5x)

b)

-35 sin(3 - 5x)

c)

-35 cos(3 - 5x)

d)

7 sin(3 - 5x)

54.

Find the derivative of the following function: g) y = 2 sin(x/4) Fill in the blank: dy/dx = _______

a)

1/2 cos(x/4)

b)

2 cos(x/4)

c)

1/4 cos(x/4)

d)

2 sin(x/4)

55.

Find the derivative of the following function: y = 4 cos(2x/5) Fill in the blank: dy/dx = _______

a)

-8/5 sin(2x/5)

b)

8/5 sin(2x/5)

c)

-4/5 sin(2x/5)

d)

-8/5 cos(2x/5)

56.

Find the derivative of the following function: i) y = tan(2x) Fill in the blank: dy/dx = _______

a)

2sec2(2x)2 sec^2(2x)

b)

sec2(2x)sec^2(2x)

c)

2 tan(2x)

d)

2cos⁡2(2x)2 \cos^2(2x)

57.

Find the derivative of the following function: j) y = 2cos⁡3(2x2)2 \cos^3(2x^2) Fill in the blank: dy/dx = _______

a)

-24x sin(2x^2) cos^2(2x^2)

b)

−12xsin⁡(2x2)cos⁡2(2x2)-12x \sin(2x^2) \cos^2(2x^2)

c)

−24xsin⁡(x2)cos⁡2(2x2)-24x \sin(x^2) \cos^2(2x^2)

d)

-24x sin(2x2)cos3(2x2)sin(2x^2) cos^3(2x^2)

58.

Find the derivative of the following function: y = 4x2+5x34x^2 + 5x^3 Fill in the blank: dy/dx = _______

a)

8x+15x28x + 15x^2

b)

4x+15x24x + 15x^2

c)

8x+5x28x + 5x^2

d)

8x2+15x8x^2 + 15x

59.

Find the derivative of the following function: y = 5x5/2−2x5x^{5/2} - 2\sqrt{x} Fill in the blank: dy/dx = _______

a)

(2/(x3/2))−(1/x)(2/ (x^{3/2})) - (1/\sqrt{x})

b)

(25/2)x3/2−(1/x)(25/2)x^{3/2} - (1/x)

c)

(5/2)x3/2−(2/x(5/2)x^{3/2} - (2/\sqrt{x}

d)

(5/2)x3/2−(2x1/2)(5/2)x^{3/2} - (2x^{1/2})

60.

Find the derivative of the following function: y = e3x−5e−2x+3e^{3x} - 5e^{-2x} + 3 Fill in the blank: dy/dx = _______

a)

3e3x+10e−2x3e^{3x} + 10e^{-2x}

b)

e3x−10e−2xe^{3x} - 10e^{-2x}

c)

3e3x−10e−2x3e^{3x} - 10e^{-2x}

d)

3e3x+5e−2x3e^{3x} + 5e^{-2x}

61.

Find the derivative of the following function: y = 5sin(2x) + (1/2)cos(5x) Fill in the blank: dy/dx = _______

a)

10cos(2x) - (5/2)sin(5x)

b)

10sin(2x) + (5/2)cos(5x)

c)

5cos(2x) + (1/2)sin(5x)

d)

10cos(2x) + (5/2)sin(5x)

62.

Find the derivative of the following function: y = (1/5)x3−3x4+6x(1/5)x^3 - 3x^4 + 6x Fill in the blank: dy/dx = _______

a)

(3/5)x2+12x3+6(3/5)x^2 + 12x^3 + 6

b)

(1/5)x2−12x3+6(1/5)x^2 - 12x^3 + 6

c)

(3/5)x2−12x3+6x(3/5)x^2 - 12x^3 + 6x

d)

(1/5)x2+12x3+6x(1/5)x^2 + 12x^3 + 6x

63.

Find the derivative of the following function: f) y = x10−5x7+4x5−2x−2+10xx^{10} - 5x^7 + 4x^5 - 2x^{-2} + 10x Fill in the blank: dy/dx = _______

a)

10x9−35x6+20x4+4x3+1010x^9 - 35x^6 + 20x^4 + \frac{4}{x^3} + 10

b)

10x9−35x6+20x4−4x3+1010x^9 - 35x^6 + 20x^4 - \frac{4}{x^3} + 10

c)

10x9−35x7+20x5+4x3+10x10x^9 - 35x^7 + 20x^5 + \frac{4}{x^3} + 10x

d)

10x9−35x6+20x4+4x3+1010x^9 - 35x^6 + 20x^4 + 4x^3 + 10

64.

Find the derivative of the following function: g) y = (5/6)e^{-3x} + 4e^{-x} Fill in the blank: dy/dx = ddx((5/6)e−3x+4e−x)\frac{d}{dx}((5/6)e^{-3x} + 4e^{-x}) = _______

a)

−(15/6)e−3x−4e−x-(15/6)e^{-3x} - 4e^{-x}

b)

−(5/6)e−3x−4e−x-(5/6)e^{-3x} - 4e^{-x}

c)

−(15/6)e−3x+4e−x-(15/6)e^{-3x} + 4e^{-x}

d)

−(5/6)e−3x+4e−x-(5/6)e^{-3x} + 4e^{-x}

65.

Find the derivative of the following function: h) y = -5sin(4x) - (1/2)cos(2x) Fill in the blank: dy/dx = _______

a)

-20cos(4x) + sin(2x)

b)

-20sin(4x) - cos(2x)

c)

-5cos(4x) - (1/2)sin(2x)

d)

-20cos(4x) - sin(2x)

66.

Find the derivative of the following function: y = √2 cos(3x) - 4sin(x/2) Fill in the blank: dy/dx = _______

a)

-3√2 sin(3x) - 2cos(x/2)

b)

3√2 sin(3x) + 2cos(x/2)

c)

-√2 sin(3x) - 4cos(x/2)

d)

-3√2 cos(3x) - 2sin(x/2)

67.

Find the derivative of the following function: b) y = 3e4x−ln⁡(4x)3e^{4x} - \ln(4x) Fill in the blank: dy/dx = _______

a)

12e4x−1/x12e^{4x} - 1/x

b)

3e4x−1/(4x)3e^{4x} - 1/(4x)

c)

4e3x−1x4e^{3x} - \frac{1}{x}

d)

12e4x−1/(4x)12e^{4x} - 1/(4x)

68.

Find the derivative of the following function: c) y = (2/3)e^{5x} + 5sinx Fill in the blank: dy/dx =

a)

(10/3)e5x+5cosx(10/3)e^{5x} + 5cosx

b)

(2/3)e5x+5sinx(2/3)e^{5x} + 5sinx

c)

(10/3)e5x+5sinx(10/3)e^{5x} + 5sinx

d)

(2/3)e5x+5cosx(2/3)e^{5x} + 5cosx

69.

Find the derivative of the following function: d) y = 3ln(6x) - (1/2)ln(8x) Fill in the blank: dy/dx = _______

a)

3/x - 1/(2x)

b)

3/x + 1/(2x)

c)

6/x - 4/x

d)

3ln(x) - (1/2)ln(x)

70.

Find the derivative dy/dx for the following: a)

a)

-3√2 sin(3x) - 2cos(x/2)

b)

-3√2 cos(3x) + 2sin(x/2)

c)

3√2 sin(3x) + 2cos(x/2)

d)

-3 sin(3x) - 2√2 cos(x/2)

71.

Find the derivative dy/dx for the following: b)

a)

12e4x−1/x12e^{4x} - 1/x

b)

4e3x+1x4e^{3x} + \frac{1}{x}

c)

8e4x+x−18e^{4x} + x^{-1}

d)

6e2x−x−16e^{2x} - x^{-1}

72.

Find the derivative dy/dx for the following: c)

a)

-10/3 e^{-5x} + ln(5)⋅sinx⋅cosxln(5)·sinx·cosx

b)

-10/3 e^{-5x} - ln(5)·sinx·cosx

c)

10/3 e^{-5x} + ln(5)⋅sinx⋅cosxln(5)·sinx·cosx

d)

-10/3 e^{-5x} + ln(5)⋅cosx⋅sinxln(5)·cosx·sinx

73.

Find the derivative dy/dx for the following: d)

a)

3/x - 1/2x

b)

3x - 1/2x

c)

3/x + 1/2x

d)

3x + 1/2x

74.

Which of the following are techniques of differentiation?

a)

Product rule

b)

Quotient rule

c)

Chain rule

d)

All of the above

75.

Fill in the blank for the product rule formula: If y = uv, then dy/dx = ___

a)

u dv/dx + v du/dx

b)

u du/dx + v dv/dx

c)

u/v + v/u

d)

du/dx + dv/dx

76.

Which of the following is NOT a step in differentiating using the product rule?

a)

Identify u and v

b)

Differentiate each part

c)

Apply the formula

d)

Integrate each part

77.

Find the derivative of the following function: y = (4x2+2)(3x3−1)(4x^2 + 2)(3x^3 - 1)

a)

dy/dx=60x4+18x2−8xdy/dx = 60x^4 + 18x^2 - 8x

b)

dy/dx=24x5+6x3−2dy/dx = 24x^5 + 6x^3 - 2

c)

dydx=12x2+9x3−1\frac{dy}{dx} = 12x^2 + 9x^3 - 1

d)

dy/dx = 12x2+18x2−8x12x^2 + 18x^2 - 8x

78.

Find the derivative of the following function: b) y = (3x2+1)2(3x^2 + 1)^2

a)

dy/dx=36x3+12xdy/dx = 36x^3 + 12x

b)

dydx=12x(3x2+1)\frac{dy}{dx} = 12x(3x^2 + 1)

c)

dydx=6x(3x2+1)\frac{dy}{dx} = 6x(3x^2 + 1)

d)

dydx=18x2+2\frac{dy}{dx} = 18x^2 + 2

79.

Find the derivative of the following function: c) y = (3x−1)e2x(3x - 1)e^{2x}

a)

dy/dx = (6x+1)e2x(6x + 1)e^{2x}

b)

dy/dx = (3x−1)2e2x(3x - 1)2e^{2x}

c)

dy/dx = (3x−1)e2x+3e2x(3x - 1)e^{2x} + 3e^{2x}

d)

dy/dx = (3x−1)e2x+2e2x(3x - 1)e^{2x} + 2e^{2x}

80.

Find the derivative of the following function: d) y = 5x3sin⁡(3x)5x^3 \sin(3x)

a)

dydx=15x2sin⁡(3x)+15x3cos⁡(3x)\frac{dy}{dx} = 15x^2 \sin(3x) + 15x^3 \cos(3x)

b)

dydx=15x2sin⁡(3x)−15x3cos⁡(3x)\frac{dy}{dx} = 15x^2 \sin(3x) - 15x^3 \cos(3x)

c)

dydx=5x2sin⁡(3x)+15x3cos⁡(3x)\frac{dy}{dx} = 5x^2 \sin(3x) + 15x^3 \cos(3x)

d)

dydx=5x3sin⁡(3x)+15x2cos⁡(3x)\frac{dy}{dx} = 5x^3 \sin(3x) + 15x^2 \cos(3x)

81.

Find the derivative of the following function: y = (4x2−3x)e5x(4x^2 - 3x)e^{5x}

a)

dy/dx = e5x(20x2−7x−3)e^{5x}(20x^2 - 7x - 3)

b)

dydx=e5x(8x−3)\frac{dy}{dx} = e^{5x}(8x - 3)

c)

dy/dx=e5x(8x2−3x)dy/dx = e^{5x}(8x^2 - 3x)

d)

dy/dx=e5x(4x2−3x+5)dy/dx = e^{5x}(4x^2 - 3x + 5)

82.

Find the derivative of the following function: f) y=(2x3+5x)y = (2x^3 + 5x)

a)

dydx=6x2+5\frac{dy}{dx} = 6x^2 + 5

b)

dydx=(6x2+5)\frac{dy}{dx} = (6x^2 + 5) cos(4x)−(8x3+20x)cos(4x) - (8x^3 + 20x) sin(4x)sin(4x)

c)

dydx=2x3+5x\frac{dy}{dx} = 2x^3 + 5x

d)

dy/dx=6x3+5xdy/dx = 6x^3 + 5x

83.

Find dy/dx of the following function: y = 4x2cos⁡(2x)4x^2 \cos(2x)

a)

dydx=8xcos⁡(2x)−8x2sin⁡(2x)\frac{dy}{dx} = 8x\cos(2x) - 8x^{2}\sin(2x)

b)

dydx=8xcos⁡(2x)+8x2sin⁡(2x)\frac{dy}{dx} = 8x\cos(2x) + 8x^2\sin(2x)

c)

dydx=4x2cos⁡(2x)−8xsin⁡(2x)\frac{dy}{dx} = 4x^2\cos(2x) - 8x\sin(2x)

d)

dy/dx=8x2cos⁡(2x)−4xsin⁡(2x)dy/dx = 8x^2\cos(2x) - 4x\sin(2x)

84.

Find dy/dx of the following function: b) y = 6x3e−x6x^3 e^{-x}

a)

dy/dx = −6x3e−x+18x2e−x-6x^3 e^{-x} + 18x^2 e^{-x}

b)

dy/dx = 6x3e−x+18x2e−x6x^3 e^{-x} + 18x^2 e^{-x}

c)

dydx=18x2e−x−6x3e−x\frac{dy}{dx} = 18x^2 e^{-x} - 6x^3 e^{-x}

d)

dydx=6x3e−x−18x2e−x\frac{dy}{dx} = 6x^3 e^{-x} - 18x^2 e^{-x}

85.

Find dy/dx of the following function: c) y = e2x[4cos⁡(3x)+6sin⁡(3x)]e^{2x}[4\cos(3x) + 6\sin(3x)]

a)

dy/dx = 26e2xcos⁡(3x)26e^{2x}\cos(3x)

b)

dy/dx = e2x[8cos⁡(3x)+18sin⁡(3x)]e^{2x}[8\cos(3x) + 18\sin(3x)]

c)

dy/dx = 2e2x[4cos⁡(3x)+6sin⁡(3x)]2e^{2x}[4\cos(3x) + 6\sin(3x)]

d)

dy/dx = e2x[12cos⁡(3x)+8sin⁡(3x)]e^{2x}[12\cos(3x) + 8\sin(3x)]

86.

Find dy/dx of the following function: d) y = 12e4x+5x2ln⁡∣3x∣\frac{1}{2}e^{4x} + 5x^2 \ln|3x|

a)

dy/dx=−2e−4x+5x+10xln⁡(3x)dy/dx = -2e^{-4x} + 5x + 10x\ln(3x)

b)

dy/dx = 2e4x+10xln⁡∣3x∣+5ln⁡∣3x∣2e^{4x} + 10x\ln|3x| + 5\ln|3x|

c)

dy/dx = 12e4x+10xln⁡∣3x∣+5x2\frac{1}{2}e^{4x} + 10x\ln|3x| + 5x^2

d)

dy/dx = 2e4x+5x2ln⁡∣3x∣+10x2e^{4x} + 5x^2\ln|3x| + 10x

87.

Find dy/dx of the following function then evaluate dy/dx when x = 0. y = (2x−3)e4x(2x - 3)e^{4x}

a)

dy/dx = -10 when x = 0

b)

dy/dx = 5 when x = 0

c)

dy/dx = -3 when x = 0

d)

dy/dx = 0 when x = 0

88.

Find dy/dx of the following function then evaluate dy/dx when x = 1. y=(3x2−5x)(x5+3x)y = (\frac{3}{x^2} - 5x)(x^5 + 3x)

a)

dy/dx = -60 when x = 1

b)

dy/dx = 0 when x = 1

c)

dy/dx = 15 when x = 1

d)

dy/dx = 30 when x = 1

89.

Find dy/dx of the following function: a) y = 5x3+x2x4+2\frac{5x^3 + x^2}{x^4 + 2}

a)

dy/dx = [(x4+2)(15x2+2x)−(5x3+x2)(4x3)](x4+2)2\frac{[(x^4 + 2)(15x^2 + 2x) - (5x^3 + x^2)(4x^3)]}{(x^4 + 2)^2}

b)

[(x4+2)(15x2+2x)+(5x3+x2)(4x3)](x4+2)2\frac{[(x^4 + 2)(15x^2 + 2x) + (5x^3 + x^2)(4x^3)]}{(x^4 + 2)^2}

c)

dydx=[(x4+2)(15x2+2x)−(5x3+x2)(4x2)](x4+2)2\frac{dy}{dx} = \frac{[(x^4 + 2)(15x^2 + 2x) - (5x^3 + x^2)(4x^2)]}{(x^4 + 2)^2}

d)

dy/dx = [(x4+2)(15x2+2x)−(5x3+x2)(4x3)](x4+2)\frac{[(x^4 + 2)(15x^2 + 2x) - (5x^3 + x^2)(4x^3)]}{(x^4 + 2)}

90.

Find dy/dx of the following function: b) y = 3x2−4x+1x2+5\frac{3x^2 - 4x + 1}{x^2 + 5}

a)

(x2+5)(6x−4)−(3x2−4x+1)(2x)(x2+5)2\frac{(x^2 + 5)(6x - 4) - (3x^2 - 4x + 1)(2x)}{(x^2 + 5)^2}

b)

dydx=[(x2+5)(6x−4)+(3x2−4x+1)(2x)](x2+5)2\frac{dy}{dx} = \frac{[(x^2 + 5)(6x - 4) + (3x^2 - 4x + 1)(2x)]}{(x^2 + 5)^2}

c)

dydx=[(x2+5)(3x2−4x+1)−(6x−4)(2x)](x2+5)2\frac{dy}{dx} = \frac{[(x^2 + 5)(3x^2 - 4x + 1) - (6x - 4)(2x)]}{(x^2 + 5)^2}

d)

6x−4x2+5\frac{6x - 4}{x^2 + 5}

91.

Find dy/dx of the following function: c) y = (3x2+4)/(x2+2)2(3x^2 + 4)/(x^2 + 2)^2

a)

dydx=[(x2+2)2(6x)−(3x2+4)(2)(x2+2)(2x)](x2+2)4\frac{dy}{dx} = \frac{[(x^2 + 2)^2(6x) - (3x^2 + 4)(2)(x^2 + 2)(2x)]}{(x^2 + 2)^4}

b)

dy/dx = [(x2+2)2(3x)−(3x2+4)(2)(x2+2)(x)](x2+2)3\frac{[(x^2 + 2)^2(3x) - (3x^2 + 4)(2)(x^2 + 2)(x)]}{(x^2 + 2)^3}

c)

dy/dx = [(x2+2)(6x)−(3x2+4)(2x)](x2+2)3\frac{[(x^2 + 2)(6x) - (3x^2 + 4)(2x)]}{(x^2 + 2)^3}

d)

[(x2+2)2(6x)+(3x2+4)(2)(x2+2)(2x)](x2+2)4\frac{[(x^2 + 2)^2(6x) + (3x^2 + 4)(2)(x^2 + 2)(2x)]}{(x^2 + 2)^4}

92.

Find dy/dx of the following function: d) y = cos x/(1 + sin x)

a)

dy/dx = [(1+sinx)(−sinx)−(cosx)(cosx)](1+sinx)2\frac{[(1 + sin x)(-sin x) - (cos x)(cos x)]}{(1 + sin x)^2}

b)

dy/dx = [(1+sinx)(cosx)+(cosx)(sinx)](1+sinx)2\frac{[(1 + sin x)(cos x) + (cos x)(sin x)]}{(1 + sin x)^2}

c)

dy/dx = [(−sinx)(1+sinx)+(cosx)2]/(1+sinx)2[(-sin x)(1 + sin x) + (cos x)^2]/(1 + sin x)^2

d)

dy/dx = [cos x - sin x]/(1 + sin x)

93.

Find dy/dx of the following function: y = e2x/sinxe^{2x}/sin x

a)

dy/dx = [sinx(2e2x)−e2x(cosx)]/(sinx)2[sin x (2e^{2x}) - e^{2x} (cos x)]/(sin x)^2

b)

dy/dx = [sinx(2e2x)+e2x(cosx)]/(sinx)2[sin x (2e^{2x}) + e^{2x} (cos x)]/(sin x)^2

c)

dy/dx = [sinx(2e2x)−e2x(sinx)]/(sinx)2[sin x (2e^{2x}) - e^{2x} (sin x)]/(sin x)^2

d)

dy/dx = [cosx(2e2x)−e2x(sinx)]/(sinx)2[cos x (2e^{2x}) - e^{2x} (sin x)]/(sin x)^2

94.

Find dy/dx of the following function: f) y = sin(4x)/(cos(4x) - sin(4x))

a)

dy/dx = [(cos(4x)−sin(4x))(4cos(4x))−sin(4x)(−4sin(4x)−4cos(4x))][cos(4x)−sin(4x)]2\frac{[(cos(4x) - sin(4x))(4cos(4x)) - sin(4x)(-4sin(4x) - 4cos(4x))]}{[cos(4x) - sin(4x)]^2}

b)

dy/dx = [(cos(4x)−sin(4x))(4sin(4x))−sin(4x)(−4cos(4x)−4sin(4x))]/[cos(4x)−sin(4x)]2[(cos(4x) - sin(4x))(4sin(4x)) - sin(4x)(-4cos(4x) - 4sin(4x))]/[cos(4x) - sin(4x)]^2

c)

dy/dx = [(cos(4x)−sin(4x))(4cos(4x))+sin(4x)(4sin(4x)+4cos(4x))]/[cos(4x)−sin(4x)]2[(cos(4x) - sin(4x))(4cos(4x)) + sin(4x)(4sin(4x) + 4cos(4x))]/[cos(4x) - sin(4x)]^2

d)

95.

Find dydx\frac{dy}{dx} for the following function: \[ y = \frac{x^4 + 2x}{x^4 + 2} \]

a)

dydx=−5x6−2x5+30x2+4x(x4+2)2\frac{dy}{dx} = \frac{-5x^6 - 2x^5 + 30x^2 + 4x}{(x^4 + 2)^2}

b)

dydx=4x3+2x4+2\frac{dy}{dx} = \frac{4x^3 + 2}{x^4 + 2}

c)

dydx=4x3+2xx4+2\frac{dy}{dx} = \frac{4x^3 + 2x}{x^4 + 2}

d)

dydx=4x3+8x(x4+2)2\frac{dy}{dx} = \frac{4x^3 + 8x}{(x^4 + 2)^2}

96.

Find dydx\frac{dy}{dx} for the following function: \[ y = \frac{x^2 + 5x}{x^2 + 5} \]

a)

dydx=4x2+28x−20(x2+5)2\frac{dy}{dx} = \frac{4x^2 + 28x - 20}{(x^2 + 5)^2}

b)

dydx=2x+5x2+5\frac{dy}{dx} = \frac{2x + 5}{x^2 + 5}

c)

dydx=2x2+5(x2+5)2\frac{dy}{dx} = \frac{2x^2 + 5}{(x^2 + 5)^2}

d)

dydx=2x2+10xx2+5\frac{dy}{dx} = \frac{2x^2 + 10x}{x^2 + 5}

97.

Find dydx\frac{dy}{dx} for the following function: \[ y = \frac{x^2 - 4x}{(x^2 + 2)^3} \]

a)

dydx=−6x3−4x(x2+2)3\frac{dy}{dx} = \frac{-6x^3 - 4x}{(x^2 + 2)^3}

b)

dydx=2x(x2+2)3−3(x2−4x)(x2+2)2(x2+2)6\frac{dy}{dx} = \frac{2x(x^2 + 2)^3 - 3(x^2 - 4x)(x^2 + 2)^2}{(x^2 + 2)^6}

c)

dydx=2x−4(x2+2)3\frac{dy}{dx} = \frac{2x - 4}{(x^2 + 2)^3}

d)

dydx=x2−4x3(x2+2)2\frac{dy}{dx} = \frac{x^2 - 4x}{3(x^2 + 2)^2}

98.

Find dydx\frac{dy}{dx} for the following function: \[ y = \arccos(1 + \sin x) \]

a)

dydx=−11+sin⁡x\frac{dy}{dx} = -\frac{1}{1 + \sin x}

b)

dydx=−11−(1+sin⁡x)2\frac{dy}{dx} = -\frac{1}{\sqrt{1 - (1 + \sin x)^2}}

c)

dydx=11−(1+sin⁡x)2\frac{dy}{dx} = \frac{1}{\sqrt{1 - (1 + \sin x)^2}}

d)

dydx=−cos⁡x\frac{dy}{dx} = -\cos x

99.

Find dydx\frac{dy}{dx} for the following function: \[ y = \frac{e^{2x}(2\sin x - \cos x)}{\sin^2 x} \]

a)

dydx=e2x(2sin⁡x−cos⁡x)sin⁡2x\frac{dy}{dx} = \frac{e^{2x}(2\sin x - \cos x)}{\sin^2 x}

b)

dydx=e2x(2cos⁡x+sin⁡x)sin⁡2x\frac{dy}{dx} = \frac{e^{2x}(2\cos x + \sin x)}{\sin^2 x}

c)

dydx=e2x(2sin⁡x+cos⁡x)sin⁡x\frac{dy}{dx} = \frac{e^{2x}(2\sin x + \cos x)}{\sin x}

d)

dydx=e2x(2sin⁡x−cos⁡x)cos⁡2x\frac{dy}{dx} = \frac{e^{2x}(2\sin x - \cos x)}{\cos^2 x}

100.

Find dydx\frac{dy}{dx} for the following function: \[ y = \frac{4}{(\cos(4x) - \sin(4x))^2} \]

a)

dydx=32(sin⁡(4x)+cos⁡(4x))(cos⁡(4x)−sin⁡(4x))3\frac{dy}{dx} = \frac{32(\sin(4x) + \cos(4x))}{(\cos(4x) - \sin(4x))^3}

b)

dydx=8(sin⁡(4x)+cos⁡(4x))(cos⁡(4x)−sin⁡(4x))3\frac{dy}{dx} = \frac{8(\sin(4x) + \cos(4x))}{(\cos(4x) - \sin(4x))^3}

c)

dydx=−32(sin⁡(4x)+cos⁡(4x))(cos⁡(4x)−sin⁡(4x))3\frac{dy}{dx} = \frac{-32(\sin(4x) + \cos(4x))}{(\cos(4x) - \sin(4x))^3}

d)

dydx=4(cos⁡(4x)−sin⁡(4x))2\frac{dy}{dx} = \frac{4}{(\cos(4x) - \sin(4x))^2}

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