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Calculus 1A, College of the Desert, Chapter 3.txt
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d/dx(a
x
) =
d/dx(a
x
)= a
x
ln a
Steps in Logarithmic Differentiation
1) Take natural logarithms of both sides of an equation y = f(x) and use the laws of Logarithms to simplify.
2) Differentiate implicitly with respect to x.
3) Solve the resulting equation for y '.
The Power Rule
F'(x) = nx
n-1
e =
e = lim(1+x)
1/x
x-->0
e =
e = lim(1+1/n)
n
x-->inf
Compressibility
is defined by introducing a minus sign and dividing this derivitive by the volume V:
isothermal compressibility = B = - 1/V dV/dP
V =
V = 5.3/P
Half-Life
broken into 3 parts
:
A) m(t) =
100e
-(ln2)t/1590
OR
m(t) = 100 • 2
-t/1590
B) mass after 1000 years
m(1000) = 100e
-(ln2)1000/1590
= aprox 65 mg
C) 100e
-(ln2)t/1590
= 30
or
e
-(ln2)t/1590
= 0.3
Continuosly compounded interest
Ao(1+r/n)
nt
Hyperbolic Identities
sinh(-x) =
sinh(-x) = -sinh(x)
Hyperbolic identities
cosh(-x) =
cosh(-x) = cosh(x)
Hyperbolic identities
cosh
2
(x)-sinh
2
(x)
cosh
2
(x) - sinh
2
(x) = 1
Hyperbolic identities
1 - tanh
2
(x) =
1 - tanh
2
(x) = sinh
2
(x)
Hyperbolic identities
sinh(x+y) =
sinh x cosh y + coshx sinhy
Hyperbolic identities
cosh(x+y) =
cosh(x+y) =
cosh x cosh y + sinh x sinh y
Derivitives of hyperbolic Id
d/ex(sinh x) =
d/dx(sinh x) = cosh x
Derivitives of hyperbolic iden
d/dx(cosh x) =
d/dx(cosh x) = sinh x
Derivitives of hyperbolic ident
d/dx(tanh x) =
d/dx(tanh x) = sech
2
x
Derivitives of hyperbolic ident
d/dx(csch x) =
d/dx(csch x) = -csch x coth x
Derivitives of hyperbolic ident
d/dx(sech x) =
d/dx(sech x) = -sech x tanh x
Derivitives of hyperbolic ident
d/dx(coth x) =
d/dx(coth x) = -csch^2 x
y = sinh
-1
x =
y = sinh
-1
x <---->
x = sinh y
y = cosh
-1
(x) =
(y = cosh
-1
(x) )<==> (cosh y = x) and y >_0
y = tanh
-1
(x) =
y = tanh
-1
(x) <==> tanh y = x
sinh
-1
(x) =
sinh
-1
(x) =
ln(x+ (x
2
+1)
1/2
X € R
cosh
-1
(x) =
cosh
-1
(x) =
ln(x + (x
2
-1)
1/2
)
X >_1
tanh
-1
(x) =
tanh-1 (x) =
(1/2)ln[(1+x)/(1-x)]
-1< x < 1
F(x) = cos x
F'(x)=
F'(x)= -sin x
F''(x) = - cos x
F'''(x) = sin x
F''''(x) = cos x
F^5(x) = - sin x
Author
Mattyj1388
ID
33031
Card Set
Calculus 1A, College of the Desert, Chapter 3.txt
Description
general info
Updated
2011-06-23T22:25:14Z
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