Master Derivatives in Calculus: Enhance Your Mechanical Engineering Skills

Question 1

Which of the following formulas represents the chain rule for differentiation?

Accepted Answer

d/dx(f(g(x))) = f'(g(x))g'(x)

Answer

d/dx(f(g(x))) = f'(x)g'(x)

Answer

d/dx(f(g(x))) = f''(x)g'(x)

Answer

d/dx(f(g(x))) = f'(x)g''(x)

Question 2

Determine the derivative of the constant function f(x) = c, where c is a constant.

Accepted Answer

0

Answer

c

Answer

1

Answer

x

Question 3

What is the geometric interpretation of the derivative of a function f(x) at a given point?

Accepted Answer

Slope of the tangent line to the curve of f(x) at that point

Answer

Rate of change of f(x) at that point

Answer

Area under the curve of f(x) up to that point

Answer

Curvature of the curve of f(x) at that point

Question 4

Consider the derivative of a function f(x), denoted as f'(x). How can you determine the intervals where f(x) is increasing or decreasing?

Accepted Answer

When f'(x) > 0, f(x) is increasing. When f'(x) < 0, f(x) is decreasing.

Answer

When f'(x) = 0, f(x) is increasing. When f'(x) ≠ 0, f(x) is decreasing.

Answer

When f'(x) > 0, f(x) is decreasing. When f'(x) < 0, f(x) is increasing.

Question 5

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

(x^2 + 1)/2x

Answer

x/(x^2 + 1)

Question 6

Find the derivative of f(x) = sqrt(x^3 + 1).

Accepted Answer

(3x^2)/(2*sqrt(x^3 + 1))

Answer

(2*sqrt(x^3 + 1))/(3x^2)

Answer

(x^3 + 1)/(2*sqrt(x))

Answer

(2*sqrt(x))/(x^3 + 1)

Question 7

Calculate the derivative of the function y = (x^2 + 1)/(x - 1).

Accepted Answer

(x^2 + 3x + 1)/(x - 1)^2

Answer

(x^2 - 3x + 1)/(x - 1)^2

Answer

(2x + 1)/(x - 1)^2

Answer

(x - 1)/(x^2 + 3x + 1)^2

Question 8

Calculate the derivative of y = arctan(x).

Accepted Answer

1/(1 + x^2)

Answer

-1/(1 + x^2)

Answer

1 - x^2

Answer

x^2 + 1

Question 9

Determine the derivative of the function f(x) = x³ - 2x² + 5x - 7.

Accepted Answer

f'(x) = 3x² - 4x + 5

Answer

f'(x) = x³ - 4x + 5

Answer

f'(x) = 3x² - 2x + 5

Answer

f'(x) = x³ - 2x + 7

Question 10

Find the derivative of the function f(x) = x³ + 2x² - 5x + 1 using the power rule.

Accepted Answer

3x² + 4x - 5

Answer

6x² + 4x - 5

Answer

3x² + 2x - 5

Answer

6x² + 2x - 5

Question 11

Evaluate the derivative of f(x) = sin(x) at x = π/4 using the chain rule.

Accepted Answer

√2/2

Answer

0

Answer

1

Answer

-√2/2

Question 12

Which of the following is the derivative of f(x) = |x|?

Accepted Answer

1 for x > 0, -1 for x < 0

Answer

1 for x ≥ 0, -1 for x < 0

Answer

0

Answer

undefined at x = 0

Question 13

Find the second derivative of f(x) = x^4 - 2x² + 3 using the power rule.

Accepted Answer

12x² - 4

Answer

12x - 4

Answer

24x - 4

Answer

12 - 4

Question 14

Find the derivative of x³ with respect to x.

Accepted Answer

3x²

Answer

x²

Answer

x⁴

Answer

3x

Question 15

What is the derivative of sin(x)?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

xcos(x)

Question 16

Which of the following is the derivative of ln(x)?

Accepted Answer

1/x

Answer

x

Answer

x²

Answer

ln(x)

Question 17

Find the derivative of e^x.

Accepted Answer

e^x

Answer

x

Answer

e^(x²)

Answer

0

Question 18

What is the derivative of the function y = 3x² + 2x - 5?

Accepted Answer

6x + 2

Answer

3x² + 2

Answer

6x - 5

Answer

3x + 2

Question 19

Which of the following represents the slope of the tangent line to the graph of a function at a given point?

Accepted Answer

The first derivative

Answer

The second derivative

Answer

The third derivative

Answer

The integral

Question 20

Find the derivative of the function g(x) = coth(x) using the hyperbolic function rules.

Accepted Answer

csch^2(x)

Answer

coth^2(x)

Answer

csch(x)

Answer

coth(x)

Question 21

Define the derivative of a function f(x).

Accepted Answer

The instantaneous rate of change of f(x) at a given point

Answer

The slope of the normal line to the graph of f(x) at a given point

Answer

The area under the curve of f(x) between two points

Answer

The integral of f(x)

Question 22

Find the derivative of the function f(x) = 2x^3 - 5x^2 + 3x - 1.

Accepted Answer

6x^2 - 10x + 3

Answer

2x^2 - 5x + 3

Answer

6x^2 - 5x + 3

Answer

2x^2 - 10x + 3

Question 23

Consider the function f(x) = sin(2x). What is the derivative of f(x)?

Accepted Answer

2cos(2x)

Answer

-2sin(2x)

Answer

cos(2x)

Answer

-cos(2x)

Question 24

Which of the following methods is used to find the derivative of a composite function?

Accepted Answer

Chain rule

Answer

Power rule

Answer

Integration

Answer

Product rule

Question 25

Find the critical points of the function f(x) = x^4 - 2x^2 + 1.

Accepted Answer

x = 0, x = ±1

Answer

x = 0, x = ±2

Answer

x = 1, x = -1

Answer

x = 2, x = -2

Question 26

Calculate the derivative of the function f(x) = e^(3x + 1).

Accepted Answer

3e^(3x + 1)

Answer

e^(3x + 1)

Answer

3xe^(3x + 1)

Answer

e^(3x)

Question 27

Which of the following is the derivative of the natural logarithmic function ln(x)?

Accepted Answer

1/x

Answer

ln(x)

Answer

x

Answer

e^(x)

Question 28

Calculate the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

f'(x) = 3x^2 + 4x - 5

Answer

f'(x) = 3x^2 + 2x - 5

Answer

f'(x) = 3x^2 - 4x + 5

Answer

f'(x) = x^2 + 4x - 5

Question 29

Determine the derivative of the function f(x) = sin(x).

Accepted Answer

f'(x) = cos(x)

Answer

f'(x) = -sin(x)

Answer

f'(x) = x * cos(x)

Answer

f'(x) = sin(x) * cos(x)

Question 30

Find the derivative of the function f(x) = e^x.

Accepted Answer

f'(x) = e^x

Answer

f'(x) = 1

Answer

f'(x) = x * e^x

Answer

f'(x) = e^(-x)

Question 31

Calculate the derivative of the function f(x) = ln(x).

Accepted Answer

f'(x) = 1/x

Answer

f'(x) = -1/x

Answer

f'(x) = ln(x)

Answer

f'(x) = x

Question 32

Calculate the derivative of the function f(x) = |x|.

Accepted Answer

f'(x) = 1 for x > 0; f'(x) = -1 for x < 0; f'(0) does not exist

Answer

f'(x) = 1

Answer

f'(x) = -1

Question 33

Find the derivative of the function f(x) = f(g(x)) where g(x) = x^2 + 1.

Accepted Answer

f'(x) = f'(g(x)) * g'(x)

Answer

f'(x) = g'(x)

Answer

f'(x) = f'(g(x))

Question 34

Calculate the derivative of the function f(x) = arctan(x).

Accepted Answer

f'(x) = 1/(1 + x^2)

Answer

f'(x) = arctan(x)

Answer

f'(x) = -1/(1 + x^2)

Question 35

Find the derivative of the function f(x) = (sin(x))^x.

Accepted Answer

f'(x) = (sin(x))^x * (ln(sin(x)) + cot(x))

Answer

f'(x) = (sin(x))^x * ln(sin(x))

Answer

f'(x) = (cos(x))^x * ln(cos(x))

Answer

f'(x) = x * (sin(x))^x

Question 36

Calculate the derivative of f(x) = x^2 + 3x - 5.

Accepted Answer

2x + 3

Answer

x^2 - 5

Answer

x^2 + 3

Answer

2x - 3

Question 37

Find the derivative of the function f(x) = e^(2x + 1).

Accepted Answer

2e^(2x + 1)

Answer

(2x + 1)e^x

Answer

e^2x + 1

Answer

x^2e^x

Question 38

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

1/(x^2 + 1)

Answer

(x^2 + 1)/x

Question 39

Find the derivative of the function f(x) = sin(3x - 2).

Accepted Answer

3cos(3x - 2)

Answer

sin(3x - 2)/3

Answer

-3sin(3x - 2)

Answer

(3x - 2)cos(3x - 2)

Question 40

Find the derivative of the function f(x) = (x^2 + 1)/(x - 2).

Accepted Answer

(2x - 4)/(x - 2)^2

Answer

(x - 2)/(x^2 + 1)^2

Answer

x^2 - 1/(x - 2)

Answer

(x^2 + 1)(-1)/(x - 2)

Question 41

Find the derivative of the function f(x) = sqrt(x^3 + 1).

Accepted Answer

(3x^2)/2sqrt(x^3 + 1)

Answer

2xsqrt(x^3 + 1)

Answer

1/(2sqrt(x^3 + 1))

Answer

x^3/2sqrt(x^3 + 1)

Question 42

Find the derivative of the function f(x) = arctan(2x).

Accepted Answer

2/(4x^2 + 1)

Answer

-2/(4x^2 + 1)

Answer

1/(2x)

Answer

(2x)/(4x^2 + 1)

Question 43

Find the derivative of the function f(x) = e^(sin(x))?

Accepted Answer

e^(sin(x))cos(x)

Answer

e^(sin(x))sin(x)

Answer

cos(x)e^(sin(x))

Answer

sin(x)e^(cos(x))

Question 44

What is the derivative of the function  f(x) = x³ + 2x² - 5?

Accepted Answer

3x² + 4x

Answer

x² + 4x

Answer

3x³ + 4x²

Answer

3x + 4

Question 45

Find the derivative of the function f(x) = sin(x) + cos(x).

Accepted Answer

cos(x) - sin(x)

Answer

-cos(x) - sin(x)

Answer

sin(x) - cos(x)

Answer

-sin(x) + cos(x)

Question 46

What is the derivative of the function f(x) = e^x?

Accepted Answer

e^x

Answer

x*e^x

Answer

e^(x-1)

Answer

-e^x

Question 47

Find the derivative of the function f(x) = ln(x).

Accepted Answer

1/x

Answer

x*ln(x)

Answer

x

Answer

-1/x

Question 48

What is the value of the derivative of the function f(x) = √x at x = 4?

Accepted Answer

1/4

Answer

1/8

Answer

4

Answer

16

Question 49

What is the partial derivative of the function f(x, y) = x²y + y³ with respect to y?

Accepted Answer

2xy + 3y²

Answer

2xy

Answer

y³

Answer

x² + y³