Calculus Differentiation: Master the Fundamentals with Interactive Quizzes

Question 1

How can the derivative of a function be used to determine its maximum and minimum values?

Accepted Answer

The derivative is zero at the points where the function has a maximum or minimum value.

Answer

The derivative is always positive at the points where the function has a maximum or minimum value.

Answer

The derivative is always negative at the points where the function has a maximum or minimum value.

Question 2

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

Accepted Answer

2e^(2x)

Answer

e^(2x)*2x

Answer

2/e^(2x)

Answer

1/e^(2x)

Question 3

Find the derivative of f(x) = (x² + 1)^5.

Accepted Answer

10(x² + 1)^4*x

Answer

5(x² + 1)^4*2x

Answer

(x² + 1)^5*2x

Answer

(x² + 1)^4*5x

Question 4

Find the derivative of f(x) = 1/x.

Accepted Answer

-1/x^2

Answer

1/x^2

Answer

-x

Answer

x

Question 5

Explain how the derivative of a function can be interpreted geometrically as the slope of the curve at a given point. Provide an example to illustrate your explanation.

Accepted Answer

The derivative at a point represents the gradient of the tangent line to the curve at that point.

Answer

The derivative is equal to the gradient of the normal line to the curve at a given point.

Answer

The concavity of the curve at a specific point determines the sign of the derivative at that point.

Answer

The area under the curve between two points is equal to the derivative at a specific point.

Question 6

Determine the derivative of f(x) = sin(2x) using the chain rule of differentiation.

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(2x)

Answer

-2cos(2x)

Question 7

State the product rule of differentiation.

Accepted Answer

(fg)' = f'g + fg'

Answer

(fg)' = f'g - fg'

Answer

(fg)' = fg' - f'g

Answer

(fg)' = fg' + fg'

Question 8

Apply the product rule of differentiation to find the derivative of f(x) = (x^2 + 3) * (x - 1).

Accepted Answer

2x^2 - 2x + 3

Answer

2x^2 + 2x - 3

Answer

2x^2 - x + 3

Answer

2x^2 + x - 3

Question 9

Describe the chain rule of differentiation.

Accepted Answer

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

Answer

(fog)' = f'(g(x)) / g'(x)

Answer

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

Answer

(fog)' = g'(f(x)) / f'(x)

Question 10

Find the derivative of f(x) = e^x using the exponential function differentiation rule.

Accepted Answer

e^x

Answer

e

Answer

x * e^x

Answer

1 / e^x

Question 11

Determine the derivative of f(x) = ln(x) using the logarithmic function differentiation rule.

Accepted Answer

1 / x

Answer

x

Answer

ln(x)

Answer

1 / ln(x)

Question 12

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

Accepted Answer

2e^(2x + 1)

Answer

2xe^(2x + 1)

Answer

e^(2x - 1)

Answer

e^(2x)

Question 13

State the chain rule of differentiation.

Accepted Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g'(x) + h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g(h'(x))

Answer

If f(x) = g(h(x)), then f'(x) = h'(g(x))