**TI86** Program file dated 06/23/04, 19:57 û ë ZTHEOR3Cë é àToD0 3KoìoëD1/D6/-DERIVITIVESoëD3/D1/-F1 GENERALLY F2 POWERS, LOGS, LNoëD5/D1/-F3 CIRCULAR FUNCTIONSF4 EXAMPLESoëD7/D1/-F5 `6EXIToLD1/-F1/3A/D2/-F2/3B/D3/-F3/3C/D4/-F4/3D/D5/4EX/3XoàQnD1 3KoàXnßoàAo7ZINITo4ZLoŽD0/D25/-DIFFERENTIATIONo6ZLINoŽD10/D0/-LET 'A' BE CONSTANT & 'x' BEoŽD16/D0/-VARIABLE: IF f(x)=Ax, THENoŽD22/D0/-f'(x)=AoŽD28/D25/-THE PRODUCT RULEoŽD34/D0/-IF f(x) & g(x) ARE DIFFERENT-oŽD40/D0/-IABLE, THEN (f*g)'=f'*g+f*g'o4ZKoLD1/4MO/4A1/D2/4PR/3T/D4/4EX/3T/D5/4QU/3QoàA1oƒo4ZLoŽD0/D0/-ie. f=2x & g=3x, THENoŽD6/D0/-(f*g)'=2*3x+2x*3oŽD12/D25/-THE QUOTIENT RULEoŽD18/D0/-(f/g)'=(f'*g-f*g')/goŽD24/D0/-ie. f=2x & g=3, THENoŽD30/D0/-(f/g)'=(2*3-2x*0)/3oŽD36/D25/-THE CHAIN RULEoŽD42/D0/-d/dx f(g(x))=f'(g(x))*g'(x)o4ZKoLD1/4MO/4A2/D2/4PR/3A/D4/4EX/3T/D5/4QU/3QoàA2oƒo4ZLoŽD0/D0/-ie. P=(2x+1), ThenoŽD6/D0/-P'=2(2x+1)*2oŽD12/D0/-DERIVITIVES OFFER AN EASY, FASToŽD18/D0/-METHOD FOR DETERMINING THE SHAPEoŽD24/D0/-OF FUNCTIONS. IF f' IS POSITIVEoŽD30/D0/-OVER AN INTERVAL, THEN f ISoŽD36/D0/-INCREASING ON THAT INTERVAL.oŽD42/D0/-OTHERWISE f IS DECREASING. IF f''o4ZKoLD1/4MO/4A5/D2/4PR/4A2/D4/4EX/3T/D5/4QU/3QoáA3oàA5oƒo4ZLoŽD0/D0/-IS POSITIVE OVER AN INTERVAL, THENoŽD6/D0/-f IS CONCAVE UP ON THAT INTERVAL.oŽD12/D0/-OTHERWISE f IS CONCAVE DOWN.o4ZKoLD2/4PR/4A3/D4/4EX/3T/D5/4QU/3QoàBo7ZINITo4ZLoŽD0/D30/-DIFFERENTIATIONo6ZLINoŽD10/D0/-If f=x^A, f'=Ax^(A-1)oŽD16/D0/-ie. f=x^3, f'=3xoŽD22/D0/-f=A^x: f'=(ln A)A^xoŽD28/D0/-f=log x, f'=1/(x ln 10)oŽD34/D0/-f=e^x, f'=e^xoŽD40/D0/-f=ln x, f'=1/xo4ZKoLD1/4MO/4B1/D2/4PR/3T/D4/4EX/3T/D5/4QU/3QoàB1oƒo4ZLoŽD0/D0/-f=e^(2x): f'=2e^(2x)oŽD6/D0/-f=e^(x^A), f'=Axe^(x^A)oŽD12/D0/-ie. d/dt e^t=2te^toŽD18/D1/-ie. d/dy (y-3y)=.5(y-3y)^.5oŽD24/D85/-*(2y-3)o4ZKoLD2/4PR/3B/D4/4EX/3T/D5/4QU/3QoàCo7ZINITo4ZLoŽD0/D25/-TRIG DERIVITIVESo6ZLINoŽD10/D0/-d/dx sin x=cos x: d/dx cos x=sin xoŽD16/D0/-d/dx tan x=1/cosx=secxoŽD22/D0/-d/dÁ sinÁ=1/(1-Á)oŽD28/D0/-d/dÁ tanÁ=1/(1+Á)oŽD34/D0/-d/dÁ cosÁ=1/(1-Á)oŽD40/D0/-d/dÁ(sin Á)=2sin Á cos Áo4ZKoLD1/4MO/4C1/D2/4PR/3T/D4/4EX/3T/D5/4QU/3QoàC1oƒo4ZLoŽD0/D0/-d/dx sinh x=cosh xoŽD6/D0/-d/dx cosh x=sinh xoŽD12/D0/-d/dx tanh x=1/(cosh x)o4ZKoLD2/4PR/3C/D4/4EX/3T/D5/4QU/3QoàDo7ZINITo4ZLoŽD0/D0/-ALWAYS CHECK YOUR DERIVITIVESoŽD6/D0/-WITH THE EQUATION EDITOR, der1oŽD12/D0/-FUNCTION, & TABLE TO MATCH DERIVSoŽD18/D0/-NUMERICALLY. SET UP THE EQ EDITOR:oŽD24/D0/-y1=problem (unSELECT)oŽD30/D0/-y2=der1(y1,x)oŽD36/D0/-y3=your answeroŽD42/D0/-(Press [TABLE] to compare results)o4ZKoLD1/4MO/4D1/D2/4PR/3T/D4/4EX/3T/D5/4QU/3QoàD1oƒo4ZLoŽD0/D0/-q(x)=(4-x)/(4+x), q'(x)=8/(4+x)oŽD6/D0/-v=(n^3+1)oŽD12/D0/-v'=(1/2)(1/(n^3+1))3n)oŽD18/D0/-=3n/(2(n^3+1))oŽD25/D0/-f=e^x, Then f'=(1/x)e^xoŽD33/D0/-d/dx x^x=(ln x+1)e^(x ln x), BECAUSEoŽD40/D25/-x^x=e^(x ln x), FOR x>0o4ZKoLD2/4PR/3D/D4/4EX/3T/D5/4QU/3QŸ"