Derivatives

Question 1

First principle

Answer 1

f’(x)=lim_h->0 f(x+h)-f(x)/h

Question 2

What is the derivative of a constant function?

Answer 2

The derivative of a constant function is 0.

Question 3

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

Answer 3

The derivative is f’(x) = n * x^(n-1).

Question 4

True or False: The derivative of f(x) = 5 is 5.

Answer 4

False.

Question 5

What is the power rule for differentiation?

Answer 5

The power rule states that d/dx[x^n] = n*x^(n-1).

Question 6

Fill in the blank: The derivative of f(x) = x^3 is ____.

Answer 6

3x^2.

Question 7

What is the derivative of f(x) = 7x?

Answer 7

The derivative is f’(x) = 7.

Question 8

True or False: The derivative of f(x) = -2x^2 is -4x.

Answer 8

True.

Question 9

What is the derivative of f(x) = x^(1/2)?

Answer 9

The derivative is f’(x) = (1/2)x^(-1/2).

Question 10

What is the product rule for differentiation?

Answer 10

If u and v are functions of x, then d/dx[uv] = u’v + uv’.

Question 11

Fill in the blank: The derivative of f(x) = x^4 * x^2 is ____.

Answer 11

6x^5.

Question 12

What is the quotient rule for differentiation?

Answer 12

If u and v are functions of x, then d/dx[u/v] = (u’v - uv’)/v^2.

Question 13

What is the derivative of f(x) = 3/x?

Answer 13

The derivative is f’(x) = -3/x^2.

Question 14

True or False: The derivative of f(x) = x^5 - 2x^3 + x is 5x^4 - 6x^2 + 1.

Answer 14

True.

Question 15

What is the derivative of f(x) = x^n + x^m?

Answer 15

The derivative is f’(x) = nx^(n-1) + mx^(m-1).

Question 16

Fill in the blank: The derivative of f(x) = 4x^3 - 3x + 2 is ____.

Answer 16

12x^2 - 3.

Question 17

What is the derivative of f(x) = x^2 + 4x + 4?

Answer 17

The derivative is f’(x) = 2x + 4.

Question 18

What is the derivative of a linear function f(x) = mx + b?

Answer 18

The derivative is f’(x) = m.

Question 19

True or False: The derivative of f(x) = x^3 + 3x^2 is 3x^2 + 6x.

Answer 19

True.

Question 20

What is the derivative of f(x) = -5x^4?

Answer 20

The derivative is f’(x) = -20x^3.

Question 21

Fill in the blank: The derivative of f(x) = 2x^2 + 3x - 1 is ____.

Answer 21

4x + 3.

Question 22

What is the derivative of f(x) = x^0?

Answer 22

The derivative is 0.

Question 23

What is the derivative of f(x) = 1/x?

Answer 23

The derivative is f’(x) = -1/x^2.

Question 24

True or False: The derivative of f(x) = 2x^3 + 5 is 6x^2.

Answer 24

False.

Question 25

What is the derivative of f(x) = 8x^2 - 4x?

Answer 25

The derivative is f'(x) = 16x - 4.

Question 26

Fill in the blank: The derivative of f(x) = x^6 is ____.

Answer 26

6x^5.

Question 27

What is the power rule for differentiation?

Answer 27

If f(x) = x^n, then f'(x) = n * x^(n-1).

Question 28

True or False: The constant multiple rule states that the derivative of c*f(x) is c*f'(x), where c is a constant.

Answer 28

True

Question 29

Fill in the blank: The chain rule states that if you have a composite function f(g(x)), then the derivative is _____.

Answer 29

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

Question 30

What is the formula for the quotient rule?

Answer 30

If f(x) = g(x) / h(x), then f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2.

Question 31

Which rule would you use to differentiate the function f(x) = (3x^2 + 2) / (5x - 1)?

Answer 31

You would use the quotient rule.

Question 32

What is the derivative of f(x) = 5x^4?

Answer 32

f'(x) = 20x^3.

Question 33

True or False: The derivative of a constant is zero.

Answer 33

True

Question 34

Short answer: What do you need to apply the chain rule?

Answer 34

You need a composite function.

Question 35

Multiple choice: Which of the following is a correct application of the power rule? A) f(x) = x^3 B) f(x) = 3x C) f(x) = sin(x)

Answer 35

A) f(x) = x^3

Question 36

Fill in the blank: For f(x) = kx^n, where k is a constant, the derivative is _____.

Answer 36

k * n * x^(n-1)

Question 37

What is the derivative of f(x) = 7?

Answer 37

f'(x) = 0.

Question 38

True or False: The chain rule can be used for functions that are not composite.

Answer 38

False

Question 39

Short answer: In the quotient rule, what does the denominator represent?

Answer 39

The square of the denominator function.

Question 40

Multiple choice: Which rule applies to f(x) = 2sin(x^2)? A) Power rule B) Chain rule C) Quotient rule

Answer 40

B) Chain rule

Question 41

What is the derivative of f(x) = 4x^5 - 3x^2 + 6?

Answer 41

f'(x) = 20x^4 - 6x.