# 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

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

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.