# Derivatives (copy)

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f’(x)

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## Tags and Description

### 44 Terms

1

f’(x)

1

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2

f’(sinx)

cosx

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3

f’(cosx)

-sinx

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4

f’(tanx)

sec^2x

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5

f’(secx)

secxtanx

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6

f’(cscx)

-cscxcotx

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7

f’(cotx)

-csc^2x

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8

f’(a^x)

a^xlna

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9

f’(e^x)

e^x

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10

f’(lnx)

1/x

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11

f’(loga(x))

1/(xlna)

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12

f’(c)

0

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13

f’(x^n)

nx^n-1

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14

f’(f(g(x)))

f’(g(x)) * g’(x)

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15

(f^-1)’(x)

1/(f’(f^-1(x))

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16

f’(sec^-1(x))

1/( |x| sqrt(x^2 -1))

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17

f’(csc^-1(x))

- 1/( |x| sqrt(x^2 -1))

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18

f’(sin^-1(x))

1/ (sqrt(1- x^2))

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19

f’(cos^-1(x))

- 1/ (sqrt(1- x^2))

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20

f’(tan^-1(x))

1 / (1+ x^2)

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21

f’(cot^-1(x))

- 1 / (1+ x^2)

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22

(d/dx) (f(x)g(x))

f’(x)g(x) + f(x)g’(x)

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23

(d/dx) (f(x)/g(x))

f’(x)g(x) - f(x)g’(x) / g(x)^2

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24

ln 1

0

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25

e^0

1

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26

∫du

u + C

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27

∫u^n (du)

(u^n+1)/n+1 + C

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28

∫ 1/u (du)

ln |u| +C

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29

∫ a^u (du)

a^u (1/ ln a) + C

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30

∫ u dv

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31

Fundamental Theory of Calculas

(d/dx) ∫ f(x) dx = f(x)

∫ f(x) dx = F(b) - F(a) where F’(x) = f(x)

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32

2nd FTOC

(d/dx) ∫ f(x) dx = f(g(x)) g’(x)

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33

Mean Value Theorem

if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b]: avg ROC = instant ROC at some point

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34

Intermediate value theorem

if f(x) is continuous and f(a)<N<f(b), f ( a ) < N < f ( b ) , the line y=N intersects the function at some point x=c. Such a number c is between a and b and has the property that f(c)=N f ( c ) = N

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35

Extreme Value Theorem

if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interva

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36

∫ k f(x) dx

k ∫ f(x) dx

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37

∫ [f(x) + g(x)]dx

∫ f(x) dx + ∫ g(x) dx

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38

∫ (a to b) f(x) dx

- ∫ (b to a) f(x) dx

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39

∫ (a to c) f(x) dx

∫ (a to b) f(x) dx + ∫ (b to c) f(x) dx

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40

∫ (a to a) f(x) dx

0

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41

f is integrable when

f is continuous over [a,b]

f is bounded on closed interval [a,b] and has at most a finite number of discontinuities

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42

selecting techniques

• long division when degree of deno <= degree of num

• completing the square when degree of deno > degree of num

• partial fractions when degree of deno > degree of num that polynomials in deno can be factored into linear non-repeating

• u-sub

• integration by parts

• improper integrals: infinite interval or unbounded integral, can converge or diverge by replacing inifite part with variable

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43

ln |0|

undefined

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44

e ^x

exponential

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