Quiz:Differentiation rules

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 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.

3 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

4 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 $\mathbb {R}$