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42 Cards in this Set
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 Back
 3rd side (hint)
Finding inverse of a function

1.) determine if function is onetoone, 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 f1(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^x1/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

