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42 Cards in this Set

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  • 3rd side (hint)
Finding inverse of a function
1.) determine if function is one-to-one, consider domain
2.) write y in place of f(x)
3.) solve for x
4.) exchange x and y, consider domain
5.) replace y with f-1(x)
d/dx sin x
cos x
d/dx cos x
-sin x
d/dx tan x
sec² x
d/dx csc x
-csc x cot x
d/dx sec x
sec x tan x
d/dx cot x
-csc² x
lim e^x
x->∞
lim e^x
x->-∞
0
lim e^x-1/x
x->0
1
eº =
1
e¹ =
e
ln e =
1
ln 1 =
0
ln e³ =
3
change of base
formula
this
equals this
∫ xⁿ dx
(x^(n + 1) / n + 1) + C
∫ e^x dx
e^x + C
∫ sin x dx
-cos x + C
∫ sec² x dx
tan x + C
∫ sec x tan x dx
sec x + C
∫ sinh x dx
cosh x + C
∫ tan x dx
ln |sec x| + C
∫ 1 / x dx
ln |x| + C
∫ a ^x dx
(a^x / ln a) + C
∫ cos x dx
sin x + C
Theorem 7 differentiating inverse functions
1 / f'(f^-1)
Find inverse value and plug into differentiated formula
d/dx (ln |x|)
1 / x
d/dx (arcsin x)
1 / √(1 - x²)

-1 < x < 1
d/dx (arccos x)
-[ 1 / √(1 - x²) ]

-1 < x < 1
d/dx (arctan x)
1 / 1 + x²
d/dx (csc^-1 x)
-[ 1 / (x)√(x² - 1) ]
d/dx (sec^-1 x)
1 / (x)√(x² - 1)
d/dx (cot^-1 x)
-[ 1 / 1 + x² ]
d/dx log base a (x)
1 / x ln(a)
d/dx a^x
a^x ln(a)
lim
x->0
(1 + x)^1/x
e
arcsin(sin x)
x for -π/2 ≦ x ≦ π/2
sin (arcsin x)
x for -1 ≦ x ≦ 1
arccos(cos x)
x for 0 ≦ x ≦ π
cos (arccos x)
x for -1 ≦ x ≦ 1