Differentiation Rules (Chapter 3) Flashcards

1
Q

What is the d/dx of ex?

A

ex

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2
Q

What is the d/dx of cos x?

A

-sin x

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3
Q

How do you find the derivative of f at x?

A

Lim as delta x -> 0: f(x + delta x) - f(x) / delta x

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3
Q

What is the d/dx of eu?

A

euu’

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4
Q

What is the d/dx of ln x?

A

1 / x

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4
Q

What is the d/dx of logax?

A

1 / (ln a) * x

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4
Q

What is the d/dx of cos-1(u)?

A

-u’ / sqrt(1-u2)

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5
Q

What is the d/dx of sec-1(u)?

A

u’ / |u| * sqrt (u2-1)

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5
Q

What is the d/dx of |u|?

A

u / |u| * (u’)

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6
Q

What is the d/dx of tan x?

A

sec2x

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7
Q

What is the d/dx of ax?

A

(ln a) * ax

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7
Q

What is the formula for volume of a conic?

A

1/3(pi)r2h

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10
Q

What is the quotient rule ([d/dx] u / v)?

A

vu’ - uv’ / v2

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11
Q

What is the alternate definition of a limit? How can this be used to determine differentiability with absolute value?

A

f(x) - f(x) / x - c

Evaluate the limit at c+ and c-. If they differ, the function is not differentiable.

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11
Q

What is the d/dx of ax?

A

(ln a) ax

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13
Q

What is the chain rule ([d/dx] f(u))?

A

f’(u) * u’ for each function in the expression.

14
Q

What is the d/dx of csc-1(u)?

A

-u’ / |u| sqrt(u2 - 1)

15
Q

What is the d/dx of au?

A

(ln a) au * u’

17
Q

What is the d/dx of sin x?

18
Q

What is the product rule ([d/dx] u * v)?

A

uv’ + vu’

20
Q

What is the simple power rule ([d/dx] xn)?

A

For xn, nxn-1.

For x1, 1.

21
Q

What is the derivative of a constant?

22
Q

What is the d/dx of sec x?

A

sec x tan x

23
Q

What is the constant multiple rule ([d/dx] c*u)?

24
What is the d/dx of csc x?
-csc x cot x
25
What is the d/dx of sin-1(u)?
u' / sqrt(1 - u2)
26
What is the d/dx of cot-1(u)?
-u' / 1 + u2
27
What is the d/dx of cot x?
-csc2x
28
What is the d/dx of logax?
1 / (ln a) x
29
What is the sum / difference rule ([d/dx] u + v / u - v?)
u' + v' / u' - v'
30
What is the d/dx of tan-1(u)?
u' / 1 + u2
32
What is the General Power Rule ([d/dx] un)?
nun-1 \* u'
33
What is the d/dx of ln u?
u' / u
33
What is the d/dx of logau?
u' / (ln a) u
35
What would you use to find the d/dx of (x-2)^2 / sqrt(x^2 + 1)?
Logarithmic differentiation. Take Ln of both sides. Use ln u for ln y. Use Ln division rule Ln (x/y) Ln x - Ln y.
36
How do you find the derivative of an inverse function?
It is the reciprocal of f'(g(x)) where g(x) is the inverse function.
37
A conical tank with vertex down is 20 feet across the top and 16 feet deep. How can we find: a. Water is flowing into the tank at a rate of 5 cubic feet per minute. How can we find the rate of change of the depth of the water when the water is 4 feet deep? b. If the water is rising at a rate of 2/3 feet per minute when h = 8, how can we determine the rate at which water is being pumped into the tank?
Given 1/3 pi r^2 h: a. Using similar triangles, find r = some value of h and back subtitute into the above formula (making sure to square and combine substituted h into h already present). r/h = 10/16, 16r = 10h, r=10/16h = 5/8h 1/3 pi (5/8h)^2 h = 1/3 pi 25/64 h^3 Then, find derivative of modified formula, being sure to use implicit differentiation for h. This is dv/dt. dv/dt = 75/192 \* pi \* h^2 \* h' = 25/64 \* pi \* h^2 \* h' Equate dv/dt with modified formula, plug in h (4) and solve for h'. 5 = 25/64 \* pi \* 4^2 \* h'; h' = 4/5pi ft/min
38
A conical tank with vertex down is 20 feet across the top and 16 feet deep. How can we find: b. If the water is rising at a rate of 2/3 feet per minute when h = 8, how can we determine the rate at which water is being pumped into the tank?
Using the formula already determined from item a, (dv/dt = 75/192 \* pi \* h^2 \* dh/dt), plug in dh/dt (2/3) and h.
39
What is Newton's method?
1. Graph function. 2. Approximate Zeros visually by guessing a value of x close to the zero. 3. Draw a table with 5 columns: n | xn | f(xn) | f**'**(xn) | xn - f(xn) / f'(xn) 4. Compute all values. For row 1, xn is your guessed value. For subsequent rows, xn is the value from column 5 in the prior row.