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

 ﻿d/dx(ax) = d/dx(ax)= ax 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) = nxn-1 e = e = lim(1+x)1/xx-->0 e = e = lim(1+1/n)nx-->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 cosh2(x)-sinh2(x) cosh2(x) - sinh2(x) = 1 Hyperbolic identities 1 - tanh2(x) = 1 - tanh2(x) = sinh2(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) = sech2 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+ (x2+1)1/2X € R cosh-1 (x) = cosh-1 (x) = ln(x + (x2-1)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 xF''(x) = - cos xF'''(x) = sin xF''''(x) = cos x F^5(x) = - sin x AuthorMattyj1388 ID33031 Card SetCalculus 1A, College of the Desert, Chapter 3.txt Descriptiongeneral info Updated2011-06-23T22:25:14Z Show Answers