# Calculus 1A, College of the Desert, Chapter 3.txt

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1. ﻿d/dx(ax) =

d/dx(ax)= ax ln a
2. 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 '.
3. The Power Rule
F'(x) = nxn-1
4. e =
• e = lim(1+x)1/x
• x-->0
5. e =
• e = lim(1+1/n)n
• x-->inf
6. Compressibility
• is defined by introducing a minus sign and dividing this derivitive by the volume V:
• isothermal compressibility = B = - 1/V dV/dP
7. V =
V = 5.3/P
8. 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
9. Continuosly compounded interest
Ao(1+r/n)nt
10. Hyperbolic Identities
sinh(-x) =
sinh(-x) = -sinh(x)
11. Hyperbolic identities
cosh(-x) =
cosh(-x) = cosh(x)
12. Hyperbolic identities
cosh2(x)-sinh2(x)
cosh2(x) - sinh2(x) = 1
13. Hyperbolic identities
1 - tanh2(x) =
1 - tanh2(x) = sinh2(x)
14. Hyperbolic identities
sinh(x+y) =
sinh x cosh y + coshx sinhy
15. Hyperbolic identities
cosh(x+y) =
• cosh(x+y) =
• cosh x cosh y + sinh x sinh y
16. Derivitives of hyperbolic Id
d/ex(sinh x) =
d/dx(sinh x) = cosh x
17. Derivitives of hyperbolic iden
d/dx(cosh x) =
d/dx(cosh x) = sinh x
18. Derivitives of hyperbolic ident
d/dx(tanh x) =
d/dx(tanh x) = sech2 x
19. Derivitives of hyperbolic ident
d/dx(csch x) =
d/dx(csch x) = -csch x coth x
20. Derivitives of hyperbolic ident
d/dx(sech x) =
d/dx(sech x) = -sech x tanh x
21. Derivitives of hyperbolic ident
d/dx(coth x) =
d/dx(coth x) = -csch^2 x
22. y = sinh-1 x =
• y = sinh-1 x <---->
• x = sinh y
23. y = cosh-1 (x) =
(y = cosh-1 (x) )<==> (cosh y = x) and y >_0
24. y = tanh-1 (x) =
y = tanh-1 (x) <==> tanh y = x
25. sinh-1 (x) =
• sinh-1 (x) =
• ln(x+ (x2+1)1/2
• X € R
26. cosh-1 (x) =
• cosh-1 (x) =
• ln(x + (x2-1)1/2)
• X >_1
27. tanh-1 (x) =
• tanh-1 (x) =
• (1/2)ln[(1+x)/(1-x)]

-1< x < 1
28. 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

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 Author: Mattyj1388 ID: 33031 Filename: Calculus 1A, College of the Desert, Chapter 3.txt Updated: 2011-06-23 22:25:14 Tags: Chapter3 Calculus 1A Folders: Description: general info Show Answers:

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