Quiz:Product rule for differentiation

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For a quiz that tests all the differentiation rules together, see Quiz:Differentiation rules.

Practical

Corresponds to Practical:Product rule for differentiation.

General difficulty level of questions in this section: School level (unless otherwise specified).

1 Suppose and are both defined and differentiable at the point 1. Suppose . What is the value of where denotes the pointwise product of functions?

42
44
54
63
The information given is insufficient to find .

2

What is the derivative of the function ? Hint for derivatives of individual functions: [SHOW MORE]

3 What is the derivative of the function for ? This question also requires use of chain rule for differentiation.

4

What is the derivative of the function for ? Hint for derivatives of individual functions: [SHOW MORE]


Formulas

General difficulty level of questions in this section: College level (unless otherwise specified)

1 Suppose and are both twice differentiable functions everywhere on . Which of the following is the correct formula for , the second derivative of the pointwise product of functions?

2 Suppose are everywhere differentiable functions from to . What is the derivative , where denotes the pointwise product of functions?


Significance

Qualitative and existential significance

Corresponds to Product rule for differentiation#Qualitative and existential significance.

General difficulty level of questions in this section: College level (unless otherwise specified).

1 Suppose and are continuous functions at and is the pointwise product of functions. Which of the following is true (see last two options!)?

If and are both left differentiable at , then so is .
If and are both right differentiable at , then so is .
If and are both differentiable at , then so is .
All of the above are true
None of the above is true

2 Suppose and are continuous functions at and is the pointwise product of functions. What is the relationship between the differentiability of , , and at ?

If any two of the three functions are differentiable at , then so is the third.
If is differentiable at , so are and .
If and are differentiable at , so is . However, differentiability of and at does not guarantee differentiability of .
If and are both differentiable at , so is . However, differentiability of and does not guarantee differentiability of , and differentiability of and does not guarantee differentiability of .
We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

3 Suppose and are continuous functions defined on all of . Suppose is the subset of comprising those points where is differentiable, and is the subset of comprising those points where is differentiable. Then, what can we say is definitely true about the subset of comprising those points where the pointwise product of functions is differentiable?

It is contained in the intersection
It contains the intersection
It is contained in the union
It contains the union
None of the above

4 Suppose and are continuous functions defined on all of . Suppose are such that . Suppose is known to be differentiable at and is known to be differentiable at . We do not have information about where else or is differentiable. What can we conclude about where the pointwise product of functions is differentiable?

is differentiable at
is differentiable at at least one of or but not necessarily at both
is differentiable at both and
We don't have enough information to conclude anything about the set of points where is differentiable

5 Suppose is a collection of differentiable functions defined on all of . Further, suppose that there is a collection of functions such that every element of can be written as a polynomial in terms of the elements of , with constant coefficients. Suppose that the derivative of every element of is in . Which of the following conditions are sufficient to ensure that the derivative of every element of is in ?

It is sufficient to ensure that is closed under addition and scalar multiplication, i.e., it forms a vector space of functions.
It is sufficient to ensure that is closed under multiplication, i.e., the product of any two elements of is in .
It is sufficient to ensure that is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), and closed under multiplication, but just having one of those conditions need not suffice.
It is not sufficient to ensure that is closed under addition-cum-scalar multiplication (i.e., it forms a vector space), and closed under multiplication


Computational feasibility significance

See the section #Practical.

Computational results significance

Corresponds to Product rule for differentiation#Computational results significance.

General difficulty level of questions in this section: College level (unless otherwise specified).

1 Suppose and are infinitely differentiable functions on all of such that both and are periodic functions with the same period . What can we conclude about ?

must be periodic
must be periodic, but may or may not be periodic.
must be periodic, but may or may not be periodic.
must be periodic, but may or may not be periodic.
We cannot conclude from the given information whether or any of the derivatives of is periodic.

2 Suppose and are functions defined and differentiable on the open interval . Suppose, further, that on , the derivative functions and are both expressible as rational functions. What can we say about and on ?

Both and are expressible as rational functions.
is expressible as a rational function, but need not be expressible as a rational function.
is expressible as a rational function, but need not be expressible as a rational function.
Neither nor need be expressible as a rational function.