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?

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

2 Suppose $f_{1},f_{2},f_{3}$ are everywhere differentiable functions from $\mathbb {R}$ to $\mathbb {R}$. 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?

	$f_{1}'\cdot f_{2}'\cdot f_{3}'$
	$f_{1}'\cdot f_{2}\cdot f_{3}+f_{1}\cdot f_{2}'\cdot f_{3}+f_{1}\cdot f_{2}\cdot f_{3}'$
	$f_{1}\cdot f_{2}'\cdot f_{3}'+f_{1}'\cdot f_{2}\cdot f_{3}'+f_{1}\cdot f_{2}\cdot f_{3}'$
	$f_{1}'\cdot f_{2}+f_{2}'\cdot f_{3}+f_{3}'\cdot f_{1}$
	$f_{1}''\cdot f_{2}'\cdot f_{3}$

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.

Computational feasibility

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