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 f and g are both defined and differentiable at the point 1. Suppose \! f(1) = 2, g(1) = 5, f'(1) = 4, g'(1) = 11. What is the value of (f \cdot g)'(1) where f \cdot g denotes the pointwise product of functions?

42
44
54
63
The information given is insufficient to find (f \cdot g)'(1).

2

What is the derivative of the function x \mapsto \exp(x) \sin x? Hint for derivatives of individual functions: [SHOW MORE]

x \mapsto \exp(x)(\sin x + \cos x)
x \mapsto \exp(x)(\cos x - \sin x)
x \mapsto \exp(x)(\sin x - \cos x)
x \mapsto \exp(x)\cos x + \exp(1) \sin x
x \mapsto \exp(x)\sin x + \exp(1) \cos x

3 What is the derivative of the function x \mapsto \sqrt{x}\sin(x^2) for x > 0? This question also requires use of chain rule for differentiation.

x \mapsto \cos(x^2)/(2\sqrt{x})
x \mapsto 2\sqrt{x}\cos(x^2)
x \mapsto 2\sqrt{x}(\cos(x^2 + \sin(x^2))
x \mapsto 2x\sqrt{x}\sin(x^2) + \cos(x^2)/(2 \sqrt{x})
x \mapsto 2x\sqrt{x}\cos(x^2) + \sin(x^2)/(2 \sqrt{x})

4

What is the derivative of the function x \mapsto x \sin x \ln x for x > 0? Hint for derivatives of individual functions: [SHOW MORE]

x \mapsto (\cos x)/x
x \mapsto (-\cos x)/x
x \mapsto \cos x \ln x + \cos x + (\sin x)/x
x \mapsto \cos x \ln x - \cos x + (\sin x)/x
x \mapsto \sin x \ln x + x \cos x \ln x + \sin x

Formulas

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

1 Suppose f and g are both twice differentiable functions everywhere on \R. Which of the following is the correct formula for (f \cdot g)'', the second derivative of the pointwise product of functions?

f'' \cdot g + f \cdot g''
f'' \cdot g + f' \cdot g' + f \cdot g''
f'' \cdot g + 2f' \cdot g' + f \cdot g''
f'' \cdot g - f' \cdot g' + f \cdot g''
f'' \cdot g - 2f' \cdot g' + f \cdot g''

2 Suppose f_1,f_2,f_3 are everywhere differentiable functions from \R to \R. What is the derivative (f_1 \cdot f_2 \cdot f_3)', where f_1 \cdot f_2 \cdot f_3 denotes the pointwise product of functions?

f_1' \cdot f_2' \cdot f_3'
f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'
f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1 \cdot f_2 \cdot f_3'
f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1
f_1'' \cdot f_2' \cdot f_3

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 f and g are continuous functions at x_0 and f \cdot g is the pointwise product of functions. Which of the following is true (see last two options!)?

If f and g are both left differentiable at x_0, then so is f \cdot g.
If f and g are both right differentiable at x_0, then so is f \cdot g.
If f and g are both differentiable at x_0, then so is f \cdot g.
All of the above are true
None of the above is true

2 Suppose f and g are continuous functions at x_0 and f \cdot g is the pointwise product of functions. What is the relationship between the differentiability of f, g, and f \cdot g at x_0?

If any two of the three functions are differentiable at x_0, then so is the third.
If f \cdot g is differentiable at x_0, so are f and g.
If f \cdot g and f are differentiable at x_0, so is g. However, differentiability of f and g at x_0 does not guarantee differentiability of f \cdot g.
If f and g are both differentiable at x_0, so is f \cdot g. However, differentiability of f \cdot g and f does not guarantee differentiability of g, and differentiability of f \cdot g and g does not guarantee differentiability of f.
We cannot draw any inferences about differentiability of one of the three functions based on differentiability of the other two.

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

It is contained in the intersection A \cap B
It contains the intersection A \cap B
It is contained in the union A \cup B
It contains the union A \cup B
None of the above

4 Suppose \mathcal{F} is a collection of differentiable functions defined on all of \R. Further, suppose that there is a collection \mathcal{B} of functions such that every element of \mathcal{F} can be written as a polynomial in terms of the elements of \mathcal{B}, with constant coefficients. Suppose that the derivative of every element of \mathcal{B} is in \mathcal{F}. Which of the following conditions are sufficient to ensure that the derivative of every element of \mathcal{F} is in \mathcal{F}.

It is sufficient to ensure that \mathcal{F} is closed under addition and scalar multiplication, i.e., it forms a vector space of functions.
It is sufficient to ensure that \mathcal{F} is closed under multiplication, i.e., the product of any two elements of \mathcal{F} is in \mathcal{F}.
It is sufficient to ensure that \mathcal{F} 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 \mathcal{F} 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 f and g are infinitely differentiable functions on all of \R such that both f' and g' are periodic functions with the same period h > 0. What can we conclude about f \cdot g?

f \cdot g must be periodic
(f \cdot g)' must be periodic, but f \cdot g may or may not be periodic.
(f \cdot g)'' must be periodic, but (f \cdot g)' may or may not be periodic.
(f \cdot g)''' must be periodic, but (f \cdot g)'' may or may not be periodic.
We cannot conclude from the given information whether any of the derivatives of f \cdot g is periodic.

2 Suppose f and g are functions defined and differentiable on the open interval (0,1). Suppose, further, that on (0,1), the derivative functions \! f' and \! g' are both expressible as rational functions. What can we say about f \cdot g and (f \cdot g)' on (0,1)?

Both f \cdot g and (f \cdot g)' are expressible as rational functions.
f \cdot g is expressible as a rational function, but (f \cdot g)' need not be expressible as a rational function.
(f \cdot g)' is expressible as a rational function, but f \cdot g need not be expressible as a rational function.
Neither f \cdot g nor (f \cdot g)' need be expressible as a rational function.