Master Differentiation: Enhance Your Calculus Skills for Electrical Engineering

Question 1

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

Accepted Answer

3x^2 - 4x + 5

Answer

x^2 - 4x + 5

Answer

3x^2 + 4x + 5

Answer

x^3 - 4x^2 + 5x + 1

Question 2

What is the derivative of the function f(x) = e^(3x)?

Accepted Answer

3e^(3x)

Answer

e^(3x)

Answer

1/e^(3x)

Answer

-e^(3x)

Question 3

Find the critical points of the function f(x) = x^3 + 6x^2 - 4x + 2.

Accepted Answer

-1, 2

Answer

1, -2

Answer

-2, 1

Answer

-1, -2

Question 4

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 5

Calculate the derivative of the function h(x) = sin(x) + cos(x).

Accepted Answer

cos(x) + sin(x)

Answer

cos(x) - sin(x)

Answer

-cos(x) - sin(x)

Answer

-cos(x) + sin(x)

Question 6

Calculate the derivative of the function f(x) = sqrt(x) + 1/sqrt(x).

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 7

Applying the power rule, find the derivative of the function f(x) = x^3:

Accepted Answer

3x^2

Answer

x^2

Answer

2x

Answer

3

Question 8

Determine the derivative of the trigonometric function f(x) = cos(x):

Accepted Answer

-sin(x)

Answer

sin(x)

Answer

-cos(x)

Answer

cos(x)

Question 9

In the first derivative test, what does a positive first derivative at a critical point indicate?

Accepted Answer

A local maximum

Answer

A local minimum

Answer

An inflection point

Answer

Neither a local minimum nor a local maximum

Question 10

Which method is commonly used to find the derivatives of functions that are defined implicitly?

Accepted Answer

Implicit differentiation

Answer

Chain rule

Answer

Quotient rule

Answer

Product rule

Question 11

Find the integral of the function f(x) = sin(x):

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

-sin(x)

Answer

sin(x)

Question 12

Calculate the derivative of y = cos(3x + 1) using the chain rule.

Accepted Answer

3sin(3x + 1)

Answer

sin(3x + 1)

Answer

-3sin(3x + 1)

Answer

-sin(3x + 1)

Question 13

Sketch the graph of the function f(x) = x^3 - 6x^2 + 9x + 2 by considering its critical points and behavior.

Accepted Answer

A downward-facing parabola with a minimum at x = 2.

Answer

An upward-facing parabola with a maximum at x = 2.

Answer

A downward-facing parabola with a maximum at x = 2.

Answer

An upward-facing parabola with a minimum at x = 2.

Question 14

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

Accepted Answer

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

Answer

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

Answer

f'(x) = 3x + 4

Answer

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

Question 15

Find the equation of the tangent line to the curve f(x) = e^x at the point (0, 1).

Accepted Answer

y = x + 1

Answer

y = e^x

Answer

y = 2x + 1

Answer

y = x

Question 16

Which of the following is NOT a fundamental differentiation rule?

Accepted Answer

Integration by parts

Answer

Quotient rule

Answer

Chain rule

Answer

Product rule

Question 17

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^3 + 4x - 5

Question 18

Which of the following correctly finds the derivative of f(x) = sin(x) using Python's sympy library?

Accepted Answer

diff(sin(x), x)

Answer

derive(sin(x))

Answer

derivative(sin(x), x)

Question 19

Sketch the graph of the function k(x) = -x^2 + 4x - 3.

Accepted Answer

A parabola opening downward with vertex at (2, 1).

Answer

A circle with center at (2, 1) and radius 3.

Answer

A parabola opening upward with vertex at (2, 1).

Answer

A line with slope -2 and y-intercept 3.

Question 20

Use logarithmic differentiation to find the derivative of the function f(x) = x^x.

Accepted Answer

f'(x) = x^x (ln(x) + 1)

Answer

f'(x) = x^(x-1)

Answer

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

Question 21

State whether the derivative of a constant function is always zero.

Accepted Answer

True

Answer

Cannot be determined

Answer

Sometimes true, sometimes false

Answer

False

Question 22

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x^2 - 5x

Answer

3x^2 + 4x + 5

Question 23

Determine the derivative of the function f(x) = e^(2x):

Accepted Answer

f'(x) = 2e^(2x)

Answer

f'(x) = 2x*e^(2x)

Answer

f'(x) = e^(x/2)

Answer

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

Question 24

Find the derivative of the function y = (x^2 + 1)/(x-1):

Accepted Answer

y' = (2x*(x-1) - (x^2 + 1))/(x-1)^2

Answer

y' = (2x + 1)/(x-1)

Answer

y' = (2x - x^2)/x-1

Answer

y' = x^2/(x-1)

Question 25

Determine the derivative of the function f(t) = sin(t) + ln(t):

Accepted Answer

f'(t) = cos(t) + 1/t

Answer

f'(t) = sin(t) + ln(t)

Answer

f'(t) = sin(t) - ln(t)

Answer

f'(t) = cos(t) - 1/t

Question 26

Which statement accurately represents the product rule of differentiation?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 27

Which statement correctly explains the Second Derivative Test?

Accepted Answer

If f''(x) > 0, x is a local minimum. If f''(x) < 0, x is a local maximum.

Answer

If f''(x) = 0, x is an inflection point.

Answer

If f'(x) = 0, x is a local maximum or minimum.

Question 28

What is the derivative of the function  *f(x) = e^(x² - sin(x))* ?

Accepted Answer

(2x - cos(x)) * e^(x² - sin(x))

Answer

e^(2x - cos(x))

Answer

(x² - sin(x)) * e^(x² - sin(x))

Answer

e^(x² - sin(x))

Question 29

Find the derivative of the function *f(x) = ln(x³ + 1)*.

Accepted Answer

(3x²)/(x³ + 1)

Answer

(x³)/(x³ + 1)

Answer

(3x)/(x³ + 1)

Answer

(1)/(x³ + 1)

Question 30

Using the first derivative test, identify the intervals where the function *f(x) = x³ - 6x² + 9x + 2* is increasing.

Accepted Answer

(0, 3)

Answer

(3, ∞)

Answer

(-∞, 3)

Answer

(-∞, 0)

Question 31

Calculate the area under the curve *f(x) = x² + 1* on the interval [0, 2].

Accepted Answer

8/3

Answer

2/3

Answer

8

Answer

3/2

Question 32

What is the derivative of the function *f(x) = 3x² - 2x + 1*?

Accepted Answer

6x - 2

Answer

9x - 2

Answer

3x - 2

Answer

6x² - 2

Question 33

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

Accepted Answer

2cos(2x)

Answer

2sin(2x)

Answer

-2cos(2x)

Answer

cos(2x)

Question 34

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

Accepted Answer

e^x

Answer

x*e^(x-1)

Answer

1/e^x

Answer

ln(x)

Question 35

Find the critical points of the function *f(x) = x³ - 3x² + 2*.

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

x = -1, x = 2

Answer

x = 0, x = 1

Question 36

Determine the slope of the tangent line to the curve y = x^3 - 2x + 1 at the point (1, 0).

Accepted Answer

m = 1

Answer

m = 2

Answer

m = -1

Answer

m = -2

Question 37

Identify the function whose derivative is always positive for all real numbers.

Accepted Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 38

Find the derivative of the function g(x) = x³ + 2x² - 5x.

Accepted Answer

g'(x) = 3x² + 4x - 5

Answer

g'(x) = 3x³ + 2x² - 5

Answer

g'(x) = x² + 4x - 5

Answer

g'(x) = x³ + 4x - 5

Question 39

Determine the derivative of the function h(x) = sin(2x).

Accepted Answer

h'(x) = 2cos(2x)

Answer

h'(x) = -2cos(2x)

Answer

h'(x) = sin(2x)

Question 40

Find the critical point of the function f(x) = x² - 4x + 3.

Accepted Answer

x = 2

Answer

x = 1

Answer

x = -2

Answer

x = 0

Question 41

Determine the local maximum of the function f(x) = -x³ + 3x² + 2.

Accepted Answer

(1, 4)

Answer

(0, 2)

Answer

(2, 4)

Answer

(-1, 4)

Question 42

Find the derivative of the inverse function y = sin⁻¹(x).

Accepted Answer

dy/dx = 1/√(1-x²)

Answer

dy/dx = √(1-x²)

Answer

dy/dx = -√(1-x²)

Answer

dy/dx = -1/√(1-x²)

Question 43

Determine the differential dy/dx for the function y = e^(2x + 1).

Accepted Answer

dy/dx = 2e^(2x + 1)

Answer

dy/dx = 2x * e^(2x + 1)

Answer

dy/dx = e^(2x + 1)

Answer

dy/dx = -e^(2x + 1)

Question 44

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

Accepted Answer

f''(x) = 6x - 4

Answer

f''(x) = 3x² - 4

Answer

f''(x) = 3x³ - 4x²

Answer

f''(x) = x³ - 4x

Question 45

Determine the equation of the tangent line to the curve y = x³ + 2x² at the point (-1, -2).

Accepted Answer

y = x³ + 2x² - 5

Answer

y = x³ + 2x² + 5

Answer

y = -x³ + 2x² - 5

Answer

y = x² + 2x - 5

Question 46

Which of the following functions is not differentiable at x = 0?

Accepted Answer

y = |x|

Answer

y = sin(x)

Answer

y = e^x

Answer

y = x²

Question 47

Find the limit of the derivative of y = (x² + 1)/(x - 1) as x approaches 1:

Accepted Answer

4

Answer

2

Answer

0

Answer

1