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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^{n1}


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


HalfLife
 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) =
 tanh1 (x) =
 (1/2)ln[(1+x)/(1x)]
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

