Quiz:Product rule for differentiation

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

For background, see product rule for differentiation and product rule for higher derivatives.

Formulas

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

	$f^{″}\cdot g+f\cdot g^{″}$
	$f^{″}\cdot g+f^{\prime }\cdot g^{\prime }+f\cdot g^{″}$
	$f^{″}\cdot g+2f^{\prime }\cdot g^{\prime }+f\cdot g^{″}$
	$f^{″}\cdot g-f^{\prime }\cdot g^{\prime }+f\cdot g^{″}$
	$f^{″}\cdot g-2f^{\prime }\cdot g^{\prime }+f\cdot g^{″}$

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

	${f_{1^{\prime }}}^{\prime }\cdot {f_{2^{\prime }}}^{\prime }\cdot {f_{3^{\prime }}}^{\prime }$
	${f_{1^{\prime }}}^{\prime }\cdot f_2\cdot f_3+f_1\cdot {f_{2^{\prime }}}^{\prime }\cdot f_3+f_1\cdot f_2\cdot {f_{3^{\prime }}}^{\prime }$
	$f_1\cdot {f_{2^{\prime }}}^{\prime }\cdot {f_{3^{\prime }}}^{\prime }+{f_{1^{\prime }}}^{\prime }\cdot f_2\cdot {f_{3^{\prime }}}^{\prime }+f_1\cdot f_2\cdot {f_{3^{\prime }}}^{\prime }$
	${f_{1^{\prime }}}^{\prime }\cdot f_2+{f_{2^{\prime }}}^{\prime }\cdot f_3+{f_{3^{\prime }}}^{\prime }\cdot f_1$
	${f_{1^{″}}}^{″}\cdot {f_{2^{\prime }}}^{\prime }\cdot f_3$

Qualitative and existential significance

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.