4.4 La Diferencial

 

Kmf`l`j`úl km kmr`vdkd

;.; Bd k`fmrmlj`db

 f   Zupcledncs uld fulj`úl  juyd kmr`vdkd ml j  mx`stm, mltcljms bd mjudj`úl km bd rmjtd tdlemltm d bd eräf`jd  j, f (j)   v`mlm kdkd pcr bd mjudj`úl yd prmv`dnmltm mstuk`dkd  f   km   ml mmbb pult pultc c  y

f '( x)  x  j 





f  (j)

Fcrnd pultc pmlk`mltm km bd rmjtd

 

Mstd mjudj`úl ms jclcj`kd jcnc dprcx`ndj`úl pcr nmk`c km uld rmjtd tdlemltm d  f    ml j  , ms kmj`r, bcs  f    y vdbcrms km  pumkml ut`b`zdrsm jcnc uld dprcx`ndj`úl km bcs vdbcrms km  nmk`dltm bd mbmjj`úl km vdbcrms km  x  nuy prúx`ncs d j . Bd prmj`s`úl km bd dprcx`ndj`úl kmpmlkmrä km qum tdl jmrjdlc sm tcnm ul vdbcr km  x   db lÿnmrc j . Bd F`eurd ;.41 numstrd qum judlkc  x  j  , mb bîn`tm km  y  ms  f (j)  .

 f  

F`eurd ;.41 Dprcx`ndj`úl km bd rmjtd tdlemltm d bd eräf`jd km

 ml

 j, f (j)   .

Momnpbc ;.41 Dprcx`ndj`úl pcr nmk`c km uld rmjtd tdlemltm Dprcx`ndr pcr nmk`c km uld rmjtd tdlemltm bd fulj`úl

 f ( x ) 

x

 ml mb pultc

(;,7)

.

Zcbuj`úl Zcbuj`úl Kmtmrn`ldlkc  Kmtmrn`ldlkc bd kmr`vdkd km bd fulj`úl ml j   ; , jclcjmncs bd pmlk`mltm km bd rmjtd tdlemltm,  f '( x) 

4 7  x  , mltcljms,

 f  '(;)   

4



7 ;

4 ; 

n  

Bd mjudj`úl km bd rmjtd tdlemltm jcl pmlk`mltm  y

4



;

( x  ;)  7 

Jcnpdrdlkc bcs vdbcrms km tmlmncs  x  f ( x) 

x

 

4 ;  y

x 4

4 ;  y qum pdsd pcr mb pultc (;,7)   ms

Dprcx`ndj`úl nmk`dltm bd rmjtd tdlemltm

   kdkcs pcr bd rmjtd tdlemltm jcl bcs vdbcrms km

 f  

  jmrjdlcs db vdbcr km  x   ;  

:.=

:.0

:.00=

:.000

;

;.114

;.11=

;.14

;.4

4.8518:

4.05;8;

4.0085;0

4.0005; 0

7

7.1117;0

7.1147;0

7.117; 0

7.17;8=

 

Kmf`l`j`úl km kmr`vdkd

 y 

4 ;

4.85=

x 4

4.05=

4.0085=

4.0005=

7

7.1117=

7.1147=

7.117=

7.17=

 

[dgbd ;.4. Jcnpdrdj`úl km

 y

 y bcs vdbcrms km

 f  

 jmrjdlcs d  x   ;    S 

 Lútmsm qum judlkc judlkc näs jmrjd sm mmljumltrm ljumltrm  x  kmb vdbcr km ;, nujhc nmocr ms bd dprcx`ndj`úl y tdl scbc jc`lj`kml judlkc  x   ; . Judlkc bd mjudj`úl  y



f '( x)  x  j 



f  (j)

Fcrnd pultc pmlk`mltm km bd rmjtd

 

 f   Zm usd jcnc uld dprcx`ndj`úl km bd eräf`jd km  , bd jdlt`kdk  x  j  sm kmlcn`ld `ljrmnmltc ml  x  y sm  y  y kmlctd jcnc  x , km ndlmrd s`n`bdr, judlkc  x  ms pmqumôc, mb jdng`c ml , kmlctdkc jcnc   pumkm dprcx`ndrsm nmk`dltm

 y  f (j x)  f (j)   f '(j) x   

Jdng`c rmdb ml y   Jdng`c dprcx`ndkc ml y

 

Kmf`l`j`úl ;.6 Bd k`fmrmlj`db  y  f ( x) Z`  ms uld fulj`úl kmr`vdgbm ml ul `ltmrvdbc dg`mrtc qum jclt`mlm d  x . Bd k`fmrmlj`db km  x ( kx ) ms  y (ky ) judbqu`mr lÿnmrc rmdb k`fmrmltm km jmrc. Bd k`fmrmlj`db km    sm kmf`lm jcnc ky  f '( x)kx

 

Mb jäbjubc km k`fmrmlj`dbms pumkm cgtmlmrsm d pdrt`r km kmf`l`j`úl dltmr`cr, jdbjubdlkc pr`nmrc bd kmr`vdkd km  f    y msjr`g`mlkc mstd nubt`pb`jdkd pcr kx  . Momnpbc ;.44 Jdbjubc km k`fmrmlj`dbms k`fmrmlj`dbms Zcbuj`úl Msjr`g`r bd k`fmrmlj`db km bds fulj`clms kdkds. Zcbuj`úl Msjr`g`r

d)

g)

Fulj`úl

Kmr`vdkd

K`fmrmlj`db

 y  x :

ky

ky  :x 7 kx

 y



 

kx

sml x

 

ky kx

j)

k)

m)

 y



x

 y  m x

 

kx  

ky kt

 

  ky

 jcs x



jcs xkx

 

  4



kx

ky

 

 y  bl t 

ky

 : x 7

 7 x  

ky 

4 7  x

ky  m x kx

 m x  



4 t   

4 ky  kt   t   

kx    

 

Kmf`l`j`úl km kmr`vdkd

 S  Z` u  y v  scl fulj`clms k`fmrmlj`dgbms km  x , mb jäbjubc sm rmdb`zd d trdvàs km bds s`eu`mltms rmebds Zund c k`fmrmlj`d?

k  u  v



ku  kv

k  ju 



jku

Qrckujtc?

k  uv 



ukv  vku

Jcj`mltm?

 u  vku  uukkv k     v7 v 

Nÿbt`pbc jclstdltm?

 

Momnpbc ;.47 Jdbjubc km k`fmrmlj`dbms k`fmrmlj`dbms Msjr`g`r bd k`fmrmlj`db km bds fulj`clms kdkds. Fulj`úl d)

g)

 y  x 7  jcs x

 y



x tdl x

Kmr`vdkd

K`fmrmlj`db

ky

ky



 7 x  sml x  kx  

ky



 x smj

ky



 47x

 

kx ky

 

kx j)

k)

 y  ; x :  :x 7

 y 

 

  x smj 7 x  tdl x  

ky

 

kx

sml x sml  x  x

 7 x  sml x

ky

 47 x 7  6 x



x  tdl x kx

 6x

 

 kx  

 



x jcs x  sml x

kt

 

7

7

x

7

ky   

 x jcs x  sml x  x 7

kx    S 

Momnpbc ;.4: K`fmrmlj`db km uld fulj`úl jcnpumstd. Jdbjubdr bd k`fmrmlj`db km Zcbuj`úl

 y  7 sm sml  ; x 

 

 f ( x )  7 sml  ; x 

Jcls`kmrdr qum

 

 f '( x)  8 jcs  ; x 

Bd kmr`vdkd ms? Bd k`fmrmlj`db ms?

ky

jcs  8 jc

 ; x  kx  S 

Momnpbc ;.4; K`fmrmlj`db km uld fulj`úl jcnpumstd.

Jdbjubdr bd k`fmrmlj`db km Zcbuj`úl

 y   ;  x 7 

Jcls cls`km `kmrdr qu qumm

:

 

 f ( x)   ;  x 7 

:

  7

Bd kmr` kmr`vd vdkd kd ms?

 f '( x)  :  ;  x 7  ( 7 x)  6 x  ;  x 7  7

Bd k`fmrmlj`db ms?

ky  6 x  ;  x 7  k kxx

7

 

Kmf`l`j`úl km kmr`vdkd

 S