Derivative Rules

Question 1

What is the derivative of a constant function?

Answer 1

The derivative of a constant function is zero.

Question 2

What is the power rule for derivatives?

Answer 2

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

Question 3

True or False: The derivative of sin(x) is cos(x).

Answer 3

True.

Question 4

What is the derivative of e^x?

Answer 4

The derivative of e^x is e^x.

Question 5

What is the product rule for derivatives?

Answer 5

If u(x) and v(x) are functions, then (u*v)’ = u’v + uv’.

Question 6

Fill in the blank: The derivative of ln(x) is ____.

Answer 6

1/x.

Question 7

What is the quotient rule for derivatives?

Answer 7

If u(x) and v(x) are functions, then (u/v)’ = (u’v - uv’)/v^2.

Question 8

What does the chain rule state?

Answer 8

If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).

Question 9

True or False: The derivative of cos(x) is -sin(x).

Answer 9

True.

Question 10

What is the derivative of tan(x)?

Answer 10

The derivative of tan(x) is sec^2(x).

Question 11

Fill in the blank: The derivative of a constant times a function is ____.

Answer 11

the constant times the derivative of the function.

Question 12

What is the derivative of x^3?

Answer 12

The derivative of x^3 is 3x^2.

Question 13

What is the second derivative?

Answer 13

The second derivative is the derivative of the first derivative.

Question 14

True or False: The derivative of a sum of functions is the sum of their derivatives.

Answer 14

True.

Question 15

What is the derivative of a function at a point?

Answer 15

It is the slope of the tangent line to the function at that point.

Question 16

What is the notation for the derivative of f with respect to x?

Answer 16

f’(x) or df/dx.

Question 17

What is the derivative of 1/x?

Answer 17

-1/x^2.

Question 18

Fill in the blank: The derivative of a polynomial function is ____.

Answer 18

the sum of the derivatives of each term.

Question 19

What is the derivative of sin^2(x)?

Answer 19

2sin(x)cos(x) or sin(2x) using the chain rule.

Question 20

True or False: The derivative of x^n is nx^(n-1) for any real number n.

Answer 20

True.

Question 21

What is the implicit differentiation?

Answer 21

A technique to find the derivative of a function when it is not explicitly defined.

Question 22

What is the derivative of a constant function k?

Answer 22

0.

Question 23

What is the derivative of sinh(x)?

Answer 23

cosh(x).

Question 24

Fill in the blank: The derivative of the inverse function f^-1(x) is ____.

Answer 24

1/f’(f^-1(x)).

Question 25

What is the derivative of log_a(x)?

Answer 25

1/(x ln(a)).

Question 26

What is a higher-order derivative?

Answer 26

A derivative taken multiple times, such as the third derivative.

Question 27

True or False: The derivative of a linear function is constant.

Answer 27

True.

Question 28

What is the derivative of sec(x)?

Answer 28

sec(x)tan(x).

Question 29

What is the formula for the Mean Value Theorem?

Answer 29

If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Question 30

What is the derivative of an exponential function a^x?

Answer 30

a^x ln(a).

Question 31

Fill in the blank: The derivative of a function at a point can be interpreted as ____.

Answer 31

the instantaneous rate of change of the function at that point.

Question 32

What is the difference quotient?

Answer 32

The expression (f(x+h) - f(x))/h as h approaches 0, used to define the derivative.

Question 33

What is the derivative of arctan(x)?

Answer 33

1/(1 + x^2).

Question 34

True or False: The derivative of a function can be negative.

Answer 34

True.

Question 35

What is the derivative of composite functions?

Answer 35

It is found using the chain rule.

Question 36

What is the derivative of x^0?

Answer 36

0, since x^0 = 1, a constant.

Question 37

Fill in the blank: The derivative of the product of two functions is given by the ____.

Answer 37

product rule.

Question 38

What does the notation f''(x) represent?

Answer 38

The second derivative of f with respect to x.

Question 39

What is the derivative of a step function?

Answer 39

The derivative is zero almost everywhere, except at points of discontinuity.

Question 40

What is the derivative of a logarithmic function?

Answer 40

The derivative of log_b(x) = 1/(x ln(b)).

Question 41

What is the derivative of cot(x)?

Answer 41

-csc^2(x).

Question 42

What is the derivative of arcsin(x)?

Answer 42

1/sqrt(1 - x^2).

Question 43

Fill in the blank: The derivative of a function can be interpreted as the ____.

Answer 43

slope of the tangent line to the graph of the function.

Question 44

What is the derivative of x^n when n is a real number?

Answer 44

nx^(n-1).

Question 45

True or False: The derivative of a function can be used to find local maxima and minima.

Answer 45

True.

Question 46

What is the derivative of an implicit function?

Answer 46

It is found using implicit differentiation.

Question 47

What is the formula for the Chain Rule?

Answer 47

If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Question 48

What is the derivative of an absolute value function?

Answer 48

It is piecewise defined; it is 1 or -1 depending on the sign of the input.

Question 49

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

Answer 49

f'(x) = 2x + 3.

Question 50

Fill in the blank: The derivative of a function at a point where it is not continuous is ____.

Answer 50

undefined.

Question 51

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

Answer 51

f'(x) = 20x^3.

Question 52

What is the derivative of the function f(x) = 3x^2 - 4x + 1?

Answer 52

f'(x) = 6x - 4.

Question 53

True or False: A function can be differentiable at a point but not continuous there.

Answer 53

False.

Question 54

What is the derivative of the function f(x) = 2x^3 + 3x^2?

Answer 54

f'(x) = 6x^2 + 6x.

Question 55

What is the derivative of the function f(x) = x^1/3?

Answer 55

f'(x) = (1/3)x^(-2/3).

Question 56

Fill in the blank: The derivative of a sum of functions is ____.

Answer 56

the sum of their derivatives.

Question 57

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

Answer 57

f'(x) = 7e^x.

Question 58

What is the derivative of the function f(x) = sin(x) + cos(x)?

Answer 58

f'(x) = cos(x) - sin(x).

Question 59

What is the derivative of a constant raised to a variable power?

Answer 59

If f(x) = a^x, then f'(x) = a^x ln(a).

Question 60

True or False: The derivative gives us information about the rate of change of a function.

Answer 60

True.

Question 61

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

Answer 61

f'(x) = 4x^3 - 4x + 1.

Question 62

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

Answer 62

f'(x) = 1/(2sqrt(x)).