Quiz:Differentiation rules

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For a full list, see Category:Differentiation rules.

Method for constructing new functions from old	In symbols	Derivative in terms of the old functions and their derivatives	Proof
pointwise sum	$f + g$ is the function $x \mapsto f(x) + g(x)$ $f_1 + f_2 + \dots + f_n$ is the function $x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)$	Sum of the derivatives of the functions being added (the derivative of the sum is the sum of the derivatives) $\! f' + g'$ $\! f_1' + f_2' + \dots + f_n'$	differentiation is linear
pointwise difference	$f - g$ is the function $x \mapsto f(x) - g(x)$	Difference of the derivatives, i.e., $f' - g'$	differentiation is linear
scalar multiple by a constant	$af$ is the function $x \mapsto af(x)$ where $a$ is a real number	$x \mapsto af'(x)$	differentiation is linear
pointwise product	$f \cdot g$ (sometimes denoted $fg$) is the function $x \mapsto f(x)g(x)$ $f_1 \cdot f_2 \cdot \dots f_n$ (sometimes denoted $f_1f_2\dots f_n$ is the function $x \mapsto f_1(x)f_2(x) \dots f_n(x)$	For two functions, $x \mapsto f'(x)g(x) + f(x)g'(x)$ For multiple functions, $x \mapsto f_1'(x)f_2(x) \dots f_n(x) + f_1(x)f_2'(x) \dots f_n(x) + \dots + f_1(x)f_2(x) \dots f_n'(x)$	product rule for differentiation
pointwise quotient	$f/g$ is the function $x \mapsto f(x)/g(x)$	$x \mapsto \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$	quotient rule for differentiation
composite of two functions	$f \circ g$ is the function $x \mapsto f(g(x))$	$x \mapsto f'(g(x))g'(x)$	chain rule for differentiation
inverse function of a one-one function	$f^{-1}$ sends $x$ to the unique $y$ such that $f(y) = x$	$\! \frac{1}{f'(f^{-1}(x))}$	inverse function theorem
piecewise definition	$f(x) := \left\lbrace \begin{array}{rl} f_1(x), & x < c \\ f_2(x), & c < x \\v, & x = c \end{array}\right.$ where $f_1, f_2$ can be extended to differentiable functions on all reals	$f' = f_1'$ to the left of $c$ and $f' = f_2'$ to the right of $c$. At $c$, $f$ is differentiable iff $f_1(c) = f_2(c) = v$ and $f_1'(c) = f_2'(c)$.	differentiation rule for piecewise definition by interval

See also Category:Differentiation rules for a list of all the differentiation rules pages (including pages for higher derivatives, which are not in this table).

Qualitative and existential questions

1 Suppose $f$ and $g$ are both functions from $\R$ to $\R$. Suppose further that $f$ and $g$ are both differentiable at a point $x_0 \in \R$. 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 $\R$ to $\R$ 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 $\R$ to $\R$ 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 $\R$?

	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 $\R$

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 $\R$. Which of the following is the correct formula for $(f \cdot g)''$, the second derivative of the pointwise product of functions?

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

3 Suppose $f$ and $g$ are both twice differentiable functions everywhere on $\R$. Which of the following is the correct formula for $(f \circ g)''$, the second derivative of the composite of two functions?

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

4 Suppose $f_1,f_2,f_3$ are everywhere differentiable functions from $\R$ to $\R$. 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?

	$f_1' \cdot f_2' \cdot f_3'$
	$f_1' \cdot f_2 \cdot f_3 + f_1 \cdot f_2' \cdot f_3 + f_1 \cdot f_2 \cdot f_3'$
	$f_1 \cdot f_2' \cdot f_3' + f_1' \cdot f_2 \cdot f_3' + f_1 \cdot f_2 \cdot f_3'$
	$f_1' \cdot f_2 + f_2' \cdot f_3 + f_3' \cdot f_1$
	$f_1'' \cdot f_2' \cdot f_3$

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

	$(f_1' \circ f_2 \circ f_3) \cdot (f_2' \circ f_3) \cdot f_3'$
	$(f_1' \cdot f_2 \cdot f_3) \circ (f_2' \cdot f_3) \circ f_3'$
	$(f_1 \circ f_2' \circ f_3') \cdot (f_2 \circ f_3') \cdot f_3$
	$(f_1 \cdot f_2' \cdot f_3') \circ (f_2 \cdot f_3') \circ f_3$
	$f_1' \circ f_2' \circ f_3'$

Predicting when things become zero

The questions here can be done in two ways. The first is to use the abstract differentiation rules to figure things out. The second is to actually determine the possibilities for the functions at hand, and then figure out what we can say about their sums, products, and composites.

1 Suppose $\! f,g$ are everywhere differentiable functions on $\R$ and $\! f' = g' = 0$ everywhere. For which of the following functions can we not conclude that the derivative is zero everywhere?

	$\! f + g$
	$f \cdot g$
	$f \circ g$
	All of the above, i.e., we cannot conclude for sure that the derivative is zero for any of these functions.
	None of the above, i.e., the derivative is necessarily zero everywhere for each of these functions

2 Suppose $\! f,g$ are everywhere twice differentiable functions on $\R$ and $\! f'' = g'' = 0$ everywhere. For which of the following functions can we not conclude that the second derivative is zero everywhere?

	$\! f + g$
	$f \cdot g$
	$f \circ g$
	All of the above, i.e., we cannot conclude for sure that the second derivative is zero for any of these functions.
	None of the above, i.e., the second derivative is necessarily zero everywhere for each of these functions

3 Suppose $\! f,g$ are everywhere thrice differentiable functions on $\R$ and $\! f''' = g''' = 0$ everywhere. For which of the following functions can we definitively conclude that the third derivative is zero everywhere?

	$\! f + g$
	$f \cdot g$
	$f \circ g$
	All of the above, i.e., the third derivative is zero everywhere for each of these functions
	None of the above, i.e., the third derivative is not necessarily zero everywhere for any of these functions