Quiz:Differentiation rules

Qualitative and existential questions

Syntax error

1 Suppose $f$ and $g$ are both functions from $\mathbb {R}$ to $\mathbb {R}$. Suppose further that $f$ and $g$ are both differentiable at a point $x_{0}\in \mathbb {R}$. Which of the following functions can we not guarantee to be differentiable at $x_{0}$?

	The sum $f+g$, i.e., the function $x\mapsto f(x)+g(x)$
	The difference $f-g$, i.e., the function $x\mapsto f(x)-g(x)$
	The product $f\cdot g$, i.e., the function $x\mapsto f(x)g(x)$
	The composite $f\circ g$, i.e., the function $x\mapsto f(g(x))$
	None of the above, i.e., they are all guaranteed to be differentiable.

2 Suppose $f$ and $g$ are both functions from $\mathbb {R}$ to $\mathbb {R}$ that are everywhere differentiable. Which of the following can we not guarantee is everywhere differentiable?

	The sum $f+g$, i.e., the function $x\mapsto f(x)+g(x)$
	The difference $f-g$, i.e., the function $x\mapsto f(x)-g(x)$
	The product $f\cdot g$, i.e., the function $x\mapsto f(x)g(x)$
	The composite $f\circ g$, i.e., the function $x\mapsto f(g(x))$
	None of the above, i.e., they are all guaranteed to be everywhere differentiable

3 Suppose $f$ and $g$ are both functions from $\mathbb {R}$ to $\mathbb {R}$ and the left hand derivatives for $f$ and $g$ exist on all of $R$. For which of the following functions can we not guarantee that the left hand derivative exists on all of $\mathbb {R}$?

4 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?

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

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

	$(f''\circ g)\cdot g''$
	$(f''\circ g)\cdot (f'\circ g')\cdot g''$
	$(f''\circ g)\cdot (f'\circ g')\cdot (f\circ g'')$
	$(f''\circ g)\cdot (g')^{2}+(f'\circ g)\cdot g''$
	$(f'\circ g')\cdot (f\circ g)+(f''\circ g'')$

6 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?

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

7 Suppose $f_{1},f_{2},f_{3}$ are everywhere differentiable functions from $\mathbb {R}$ to $\mathbb {R}$. What is the derivative $(f_{1}\circ f_{2}\circ f_{3})'$ where $\circ$ denotes the composite of two functions? In other words, $(f_{1}\circ f_{2}\circ f_{3})(x):=f_{1}(f_{2}(f_{3}(x)))$.

	$(f_{1}'\circ f_{2}\circ f_{3})\cdot (f_{2}'\circ f_{3})\cdot f_{3}'$
	$(f_{1}'\cdot f_{2}\cdot f_{3})\circ (f_{2}'\cdot f_{3})\circ f_{3}'$
	$(f_{1}\circ f_{2}'\circ f_{3}')\cdot (f_{2}\circ f_{3}')\cdot f_{3}$
	$(f_{1}\cdot f_{2}'\cdot f_{3}')\circ (f_{2}\cdot f_{3}')\circ f_{3}$