MATH 100 & 180 Flashcards: Derivatives and Key Formulas

Question 1

$\dfrac{d}{dx}\,x^a$

Accepted Answer

$a\,x^{a-1}$
Works for any constant $a$. Rewrite roots and fractions as powers first.

Question 2

$\dfrac{d}{dx}\,e^x$

Accepted Answer

$e^x$

Question 3

$\dfrac{d}{dx}\,a^x$ (like $2^x$)

Accepted Answer

$a^x \log a$
Not $x\,a^{x-1}$: the variable is in the exponent, so this is not the power rule.

Question 4

$\dfrac{d}{dx}\,\log x$

Accepted Answer

$\dfrac{1}{x}$
In this course $\log$ means the natural log (base $e$), the key marked "ln" on a calculator.

Question 5

$\dfrac{d}{dx}\,\sin x$

Accepted Answer

$\cos x$

Question 6

$\dfrac{d}{dx}\,\cos x$

Accepted Answer

$-\sin x$
The minus sign is the one people forget.

Question 7

$\dfrac{d}{dx}\,\tan x$

Accepted Answer

$\sec^2 x$

Question 8

$\dfrac{d}{dx}\,\arcsin x$

Accepted Answer

$\dfrac{1}{\sqrt{1 - x^2}}$

Question 9

$\dfrac{d}{dx}\,\arctan x$

Accepted Answer

$\dfrac{1}{1 + x^2}$

Question 10

Product rule: $(fg)'$

Accepted Answer

$f'g + fg'$

Question 11

Quotient rule: $\left(\dfrac{f}{g}\right)'$

Accepted Answer

$\dfrac{f'g - fg'}{g^2}$
“Low d-high minus high d-low, over low squared.”

Question 12

Chain rule: $\dfrac{d}{dx}\,f(g(x))$

Accepted Answer

$f'(g(x))\cdot g'(x)$
Derivative of the outside, inside left alone, times the derivative of the inside.

Question 13

Definition of the derivative

Accepted Answer

$f'(x) = \displaystyle\lim_{h \to 0}\dfrac{f(x + h) - f(x)}{h}$
If a question says “use the definition”, this limit is the only method that gets marks.

Question 14

$\dfrac{d}{dx}\,\log(u)$

Accepted Answer

$\dfrac{u'}{u}$

Question 15

$\dfrac{d}{dx}\,\log y$ ($y$ depends on $x$)

Accepted Answer

$\dfrac{y'}{y}$
This is the step at the heart of logarithmic differentiation.

Question 16

$\dfrac{d}{dx}\,y^2$

Accepted Answer

$2y\,y'$
Every time you differentiate a $y$, multiply by $y'$.

Question 17

$\dfrac{d}{dx}\,(xy)$

Accepted Answer

$y + x\,y'$
It is a product, so use the product rule.

Question 18

$\dfrac{d}{dx}\,e^{y}$

Accepted Answer

$e^{y}\,y'$

Question 19

$\log(ab)$, $\ \log\dfrac{a}{b}$, $\ \log(a^k)$

Accepted Answer

$\log(ab) = \log a + \log b$
$\log\dfrac{a}{b} = \log a - \log b$
$\log(a^k) = k\log a$
There is no rule for $\log(a + b)$.

Question 20

Point-slope form of a line

Accepted Answer

$y - y_1 = m(x - x_1)$

Question 21

Slope of the tangent line to $y = f(x)$ at $x = a$

Accepted Answer

$f'(a)$

Question 22

Is the derivative $f'(x)$ the tangent line?

Accepted Answer

No. $f'(a)$ is only the slope of the tangent line.
The tangent line is $y - f(a) = f'(a)(x - a)$
A very common mistake: the derivative gives one number at $x = a$, the slope. The tangent line is a whole line, so it needs the point too.

Question 23

Tangent line at $x = a$

Accepted Answer

$y - f(a) = f'(a)(x - a)$

Question 24

Slope of a line parallel / perpendicular to slope $m$

Accepted Answer

Parallel: $m$
Perpendicular: $-\dfrac{1}{m}$

Question 25

Horizontal tangent? Vertical tangent?

Accepted Answer

Horizontal: $\dfrac{dy}{dx} = 0$
Vertical: the bottom of $\dfrac{dy}{dx}$ is $0$ (top not $0$)

Question 26

Linear approximation at $x = a$

Accepted Answer

$L(x) = f(a) + f'(a)(x - a)$
It is exactly the equation of the tangent line at $x = a$, $y - f(a) = f'(a)(x - a)$, solved for $y$.

Question 27

Average rate of change on $[a, b]$

Accepted Answer

$\dfrac{f(b) - f(a)}{b - a}$
The slope of the line through the two endpoints. Average velocity is the same thing.

Question 28

$\displaystyle\lim_{x \to 0}\dfrac{\sin x}{x}$

Accepted Answer

$1$

Question 29

Who wins as $x \to \infty$: $\log x$, $x^p$, $e^x$?

Accepted Answer

$\log x \ll x^p \ll e^x$ (any power $p > 0$)
Use it on terms that are added or subtracted: keep the winner, drop the rest. Pieces that are multiplied or divided are never dropped.

Question 30

How do you get rid of a $\log$? How do you get rid of $e$?

Accepted Answer

A log: raise $e$ to the power of both sides: $\log x = 3 \Rightarrow x = e^{3}$
An e: take the $\log$ of both sides: $e^{x} = 5 \Rightarrow x = \log 5$
$\log$ and $e$ undo each other: $e^{\log x} = x$ and $\log(e^{x}) = x$.

Question 31

$\sqrt{x^2}$

Accepted Answer

$|x|$
So it equals $-x$ for negative $x$, which matters as $x \to -\infty$.

Question 32

Where is a vertical asymptote?

Accepted Answer

Where the bottom $= 0$ and the top $\neq 0$
Write it as $x = a$
If the top is $0$ too, cancel first: it may be a hole instead.

Question 33

Where is a horizontal asymptote? (polynomial over polynomial)

Accepted Answer

Top power smaller: $y = 0$
Same power: $y = \dfrac{\text{top number}}{\text{bottom number}}$
Top power bigger: none
In general: keep the dominant term on top and bottom and cancel ($e^x$ beats any power). Write the answer as $y = L$.

Question 34

$f'(x) > 0$ on an interval means…

Accepted Answer

$f$ is increasing there
(the graph goes up, left to right)

Question 35

$f'(x) < 0$ on an interval means…

Accepted Answer

$f$ is decreasing there
(the graph goes down, left to right)

Question 36

$f''(x) > 0$ on an interval means…

Accepted Answer

$f$ is concave up there
(it bends like a cup, $\cup$)
The slope $f'$ is increasing.

Question 37

$f''(x) < 0$ on an interval means…

Accepted Answer

$f$ is concave down there
(it bends like a cap, $\cap$)
The slope $f'$ is decreasing.

Question 38

Critical point

Accepted Answer

$f'(x) = 0$
or $f'(x)$ does not exist
(at a point where $f$ is defined)

Question 39

At a critical point, $f'$ changes sign. $+$ to $-$? $-$ to $+$?

Accepted Answer

$+$ to $-$: local max
$-$ to $+$: local min
No sign change: neither
Up then down is a hilltop; down then up is a valley.

Question 40

$f'(a) = 0$. What does $f''(a) > 0$ say? $f''(a) < 0$?

Accepted Answer

$f''(a) > 0$: local min (a cup)
$f''(a) < 0$: local max (a cap)
$f''(a) = 0$: the test says nothing
When $f''(a) = 0$, go back to the sign of $f'$ on each side.

Question 41

Inflection point

Accepted Answer

Where $f''$ changes sign.
$f'' = 0$ alone is not enough.

Question 42

Here is $f$. Where is $f'$ negative?

Accepted Answer

$-1 < x < 1$
where $f$ goes down

Question 43

Here is $f$. Where is $f''$ positive?

Accepted Answer

$x > 0$
where $f$ is concave up ($\cup$)
At $x = 0$ the bend changes from $\cap$ to $\cup$: an inflection point.

Question 44

This is the graph of $f'$, not $f$. Where is $f$ increasing?

Accepted Answer

$x < -1$ and $x > 1$
where $f'$ is above the axis
Read the sign of $f'$ (above or below the axis), not whether its graph goes up.

Question 45

This is the graph of $f'$. Where does $f$ have a local max? A local min?

Accepted Answer

Max at $x = -1$ ($f'$: $+$ to $-$)
Min at $x = 1$ ($f'$: $-$ to $+$)

Question 46

This is the graph of $f'$. Where is $f$ concave up?

Accepted Answer

$x > 0$
where $f'$ is increasing (its graph goes up)
$f''$ is the slope of $f'$, so concave up is where $f'$ rises.

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