Quiz:Differentiation rules

Qualitative and existential questions

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 $\mathbb{R}$?

	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 have a left hand derivative on all of $\mathbb{R}$

Generic point computation questions

1 Which of the following verbal statements is not valid as a general rule?

	The derivative of the sum of two functions is the sum of the derivatives of the functions.
	The derivative of the difference of two functions is the difference of the derivatives of the functions.
	The derivative of a constant times a function is the same constant times the derivative of the function.
	The derivative of the product of two functions is the product of the derivatives of the functions.
	None of the above, i.e., they are all valid as general rules.

2 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?

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

3 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?

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

4 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?

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

5 Suppose $f_1,f_2,f_3$ are everywhere differentiable functions from $\mathbb{R}$ to $\mathbb{R}$. What is the derivative $(f_1\circ f_2\circ f_3{)}^{\prime }$ where $\circ$ denotes the composite of two functions? In other words, $(f_1\circ f_2\circ f_3)(x):=f_1(f_2(f_3(x)))$.

	$({f_{1^{\prime }}}^{\prime }\circ f_2\circ f_3)\cdot ({f_{2^{\prime }}}^{\prime }\circ f_3)\cdot {f_{3^{\prime }}}^{\prime }$
	$({f_{1^{\prime }}}^{\prime }\cdot f_2\cdot f_3)\circ ({f_{2^{\prime }}}^{\prime }\cdot f_3)\circ {f_{3^{\prime }}}^{\prime }$
	$(f_1\circ {f_{2^{\prime }}}^{\prime }\circ {f_{3^{\prime }}}^{\prime })\cdot (f_2\circ {f_{3^{\prime }}}^{\prime })\cdot f_3$
	$(f_1\cdot {f_{2^{\prime }}}^{\prime }\cdot {f_{3^{\prime }}}^{\prime })\circ (f_2\cdot {f_{3^{\prime }}}^{\prime })\circ f_3$