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?

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