Definisi Derivative

Proses menemukan/mendapatkan derivative/turunan dari sebuah fungsi disebut diferensiasi (diferensial).

Deivative dari fungsi f(x) :

r(x)=limf(*+h - /

h^o        h

atau

f'(-)=lim f (x" f (')

Ax ^ 0

2

 

Notasi Derivative

Jika fungsi y = f(x) derivative-nya dinyatakan

f'(Xo), ^^, D(f )(x0), Df (x0)

dx

2

dy d dx dx

f (x), yf'(x).

3

f '(x ) = y ' =

dy = df dx dx

d dx

f (x ) = Df (x ) = Dxf (x)

 

1

Contoh

1

Tentukan derivative dari fungsi f(x) = x2 - x

f.(x) = Lim f(x + h) -f (x) = Lin 2*+>rj±

h^0 h h ^0 h

= Lim (x + h) - (x + h) - (^ - x) = Lim(2x + h -1)

h^0 h h^0

2 2 2

T. x + 2xh + h -x-h-x + x 01

= Lim- = 2x - 1

h^0 h

 

Contoh

2

Tentukan derivative dari fungsi f(x) = V3x + 2

f'(x) = Limf (x + h - f (x)

J              h ^o       h

J3(x + h) + 2 -V3x + 2 = Lim —-

h^O       h

= Lim

h^O

^3(x + h) + 2 -43x + 2 ^3(x + h) + 2 + V3x + 2 h     t]3(x + h) + 2 W3x + 2

= Lim

3 x + 3h + 2 - 3 x - 2

= Lim-

3

h^0 h(^3(x + h) + 2 W3x + 2)       ^3(x+h)+2 W3x+2)

3h

L/m-.     =-,

hh^3(x + h) + 2 + V3x + 2)

3

2(V3 x + 2)

 

Grafik Derivative dari Fungsi

Jika f(x) naik (slope positif

Jika f(x) turun (Slope negatif

Tangen horizontal (slope =0)

 

Derivative Fungsi Tidak Ada

Sudut

Tangen vertikal Diskontinue

 

Aturan Dasar Diferensiasi

2

3

4

5

d

(c) = 0    Konstanta

dx

d d

--x(cu) = cdf(u) Perkalian konstanta

CUv        CUv

D (x) = 1               Derivative f(x)=x adalah 1.

D(f + g) = D(f) + D(g) atau d( + v) = d(u) + d(v)

CtA         OlJv        CtA

Penjumlahan dan perbedaan

D (fg ) = D (f )g+ fD (g) atau -d- (uv ) = ud (v) + v—(u)

dx           dx           dx

Produk

1

 

6

d lxn ) = nxn-1 Pangkat

dx

7

D (f (g (x))) = D (f)(g (x)) D (g)( x)              Rantai

=f '(g (x ))g'(x)

dx

8

D

' n gD(f)-fD(g) atau A

v g J "

 

g2           dx V v J

vd (u) - u—(v) dx             dx

2

v

Hasi bagi

9

1

D(f-1 ) = D(f) Funfsi Invers

 

Contoh

Konstanta

f (x) = 5 f'(x) = 0

2

3

f (x) = 3 x

8

f'(x) = 3 (8 x7 ) = 24 x7

f ( x ) =

4t

2

5

ff ( x ) =

d_ dx

4 x

4 d

5 dx

x

2

= 4 (2 x )=

5              5

 

1

Contoh

f (x) =

x

.2

1

34~x

+17

1

1 2 1 -

f (x) = — x — x 2 +17 5 3

f'(x) = 2 *1 x + (- - x" 2 + 0

5 3 2

2 1

f f( x)=5 x+—

3

6 x 2

1

 

2

f (x) = x3 - 4 x + 5

f'(x) = —x3 - — 4x + — 5

dx dx dx

f' (x) = 3 x2 - 4 + 0

f' (x) = 3 x2 - 4

3

11 11

f' (x) = 0 +12 x 1 = 12 x f (x) = 7 + x

 

Contoh

f(x) = x

7

fix) = Ix

 

 

x +4x

/'(*) = -( 3x2+4x

3x + 2

V5

x +4x

xl/2

= (3x2 + 4xj )"1/2(6x + 4)

 

Contoh

y = (3x2 - 2x_1)(4x3 + 5)

^ = (3x2 - 2x-1) -d- (4x3 + 5) + (4x3 + 5)—(3x2 - 2x)

dx           dx           dx

dx

= (3xz - 2x)(12xz) + (4xJ + 5)(6x + 2x-)

dy = 36x4 - 24x + 24x4 + 8x + 30x +10x"2

dx

dy r^ . 1 . 1

dx

= 60 x4 +14 x +10 x

1

 

2

3

f (x) = (3 x - 2 x 2)(5 + 4 x)

f' (x) = (3 x - 2 x2)—(5 + 4x) + (5 + 4 x)—(3 x - 2 x2)

dx           dx

= (3 x - 2 x 2)4 + (5 + 4 x )(3 - 4 x) = -24 x2 + 4 x +15

h (x) = (3 x - 2 x 2)(5 + 4 x).

f (x) = 3 x - 2 x2 ^ f' (x) = 3 - 4 x

g (x) = 5 + 4 x ^ g' (x) = 4

(fg )'= fg + fg ' •• h' (x) = fg + fg'

h' (x) = (3 - 4 x )(5 + 4 x) + (3 x - 2 x2 )(4) h ' (x) = -24 x2 + 4 x +15

 

Contoh

d             r f (x )i   g (x) dd- [f (x) ]- f (x) d [g (x) ]

dx           _ g(x) _ [g ( x ) ]2

y =

dy dx

5 x - 2

x2 +1

(x2 + 1)

2 ? 1 — (5x - 2) - (5x - 2) — (x2 +1)

dx           dx

(x2 +1)2

dy = (x2 +1)(5) - (5 x - 2)(2 x) dx (x2 +1)2

dy = (5x2 + 5) - (10x2 - 4x) dx      (x2 +1)2

dy -5 x2 + 4 x + 5 dx (x2 +1)2

1

 

2

f ( x ) =

5 x - 2 x2 +1

f' (x) =

(x2 +1)—(5 x - 2) - (5 x - 2)—(x2 +1) _dx_dx_

(x2 +1)2

(x2 +1)5 - (5x - 2)2x (5x2 + 5) - (10x2 - 4x)

(x2 +1)2

(x2 +1)2

-5 x + 4 x + 5

(x2 +1)2

3

h( x) =

5 x - 2

x2 +1 f (x) = 5 x - 2

g (x) = x +1

f' (x) = 5

g' (x) = 2 x

f_

g

f g

= (5)( x2 +1) - (5 x - 2)(2 x) h (x) = (x2 +1)2

= fg - fg'

" [g]2

-5 x2 + 4 x + 5 (x2 +1)2

 

4

f (x) = ^ x2 + 1

(x2 +1)d (5 x - 2) - (5 x - 2)d (x2 +1) f' ( x ) =_dx_dx_

(x2 +1)2

(x2 +1)5 - (5x - 2)2x = (5x2 + 5) - (10x2 - 4x) _ -5x2+4x + 5 (x2 +1)2 = (x2 +1)2 _ (x2+l)2

 

Contoh

Rantai

2

d dx

[f( g (x)) ] = f' (g (x)) g ' ( x)

r

G (x) =

2 x -1 ^

V

3 x + 5

 

r

G' (x) = 7

V

2              x -1

3              x + 5

r

\

V

r

G' (x) = 7

2 x -1

(3 x + 5 )2-(2 x -1)3

2

(3 x + 5) 13 91(2 x -1)

J

V

3x + 5 J (3x + 5)2 (3x + 5)

8

y = u5//2, u = 7x8 + 3x1 = 2u32 -(56x7 + 6x)

= 2(7x8 + 3x2 ) -(56x7 + 6x) = (140x7 +15x)(7x8 + 3x2)

\3/2

1

 

3

f (x) = (3 x - 5 x2)7

4

f (x) = 3(x2 -1)2

u =

y =

2 du x -1 ^ = 2 x

dx

2 dy 2 -3 u3 ^ =—u 3

du 3

dy = 7u 6 3 -10 x) dx

dy = 7(3x - 2 x 2)6(3 -10 x)

dx

dy dx

dy

dx dy

dx

3 u 3 ^ 2 x)

2

1

-(x2 -1) 32x) 4 x

1

3(x2 -1)3

 

Aturan Fungsi Khusus

2

3

4

5

dxr r-1 -= rx ', r e ?

dx

d sin (x)

dx

= cos(x)

dcos(x)

dx

= - sin (x)

dtan(x) 1

dx cos2(x) d arcsin (x) 1

dx

4T-

x

6

7

8

9

d arctan (x) 1

dx

1 + x

2

d ex

dx

= e

x

dax dx

= ax ln (a)

d ln (x) = 1 dx x

1

 

Con+oh

71 / (*) = sin(2x)

. du u = 2x^ — = 2

dx

y = sin(w) —» — = cos (u)

du

f (x) = tan(x -1)

? du -u-x -1 —> — = 2x

dx

2, x

y = tan(tt) ^ — = sec (w)

du

— = 2 cos (u) = 2cos(2x) dx

— = sec2(^)(2x) = 2xsec2(x2 -1) dx

 

Diferensiasi Fungsi Sinus

d sin (x)

dx

= cos(x)

sin(x + h) - sin(x) sin(x)cos(h) + cos(x)sin (h) - sin (x)

h h

sin(h) / . / .cos(h)-1 = —cos (x) + sin (x)-——

h w w h

sin(x + h)-sin(x) , . ^ lim—----= cos(x) : incat bahwa :

h^o        h             v ;

sin (h) , cos (h)-1 ^ lim—= 1         aan         lim-^— = 0 v

h^o h    h^o h

 

Diferensiasi Fungsi Cosinus

dcos(x) . ( . -±-L = - sin (x)

dx

( n           >                             (n 1

                - x           = cos      --x 1

V 2          J                              V 2 J

r

v

n

--x

2

A

J

= - sin (x)

 

Diferensiasi Fungsi Tangen

d tan(x) 1

dx

d tan (x)

cos2(x)

d

( sin(      x)            >

V cos     (x            )J

dx

dx

cosi        (x)          cos         (x)          !-sin (                    >)            (-sini      (x))

cos2(                                                                     (x;           )                             

cos2       (x)          1 + sin2!                               M            i = 1

cos2(x)                                 1 cos2(x)                             

 

Diferensiasi Inverse Fungsi Trigonometri

d arcsin (x) 1

dx VTT-2

x

Jika x = sin (y ).  >,

d arcsin (x)          1111

dx           d (sin (y)) cos (y) - sin2 (y) V1-r

dy

2

d arctan (x) 1

dx           1 + x2

Jika x = tan (y)

darctan(x) 1 2/ . 1 1 _i_l =_= cos2 (y) =_=_

dx d tan(y) v 7 1 + tan2 (y) 1 + x2

dy

1

 

Diferensiasi Fungsi Exponensial

d ax

dx

= ax

ax+h - ax xah -1 x ah - a0               x

-= ax-= ax--z 0 ^ ax

h h h h^0

ah - a0

Ingat dafinisi dari bilangan naturan a lim-= 1

h

2

da

x

= ax ln(a)

dx

i-i            x ln(ax) xln(a)

Jika a = a v ; = a w dan nuanggunakan aturan rantai Maaa

d (ax ln(a)

da

= axln(a) ln (a) = ax ln (a).

dx dx

 

3

A [[ (x )]= limi^HM

dx           h ^0 h

— (ex )= limiX+l^il

dx v 7 h ^ 0 h

h x

e e -e lim-

h ^0 h

lim -e—^-1

h ^0 h

eh -1

exlim-= ex(1) = e

h^o h

 

Aturan Diferensiasi Fungsi Exponensial

Aturan 1:

d /

dx

(e" )=

xl ex

Aturan 2:

d /

dx

(ef (x) )= ef (x) • f'(x)

 

Contoh

Temukan derivative dari f(x) = x2ax

f(x) = x2ex f '(x) = x2ex + ex2x

f '(x) = xex (x + 2)

2

Temukan derivative dari f(t) = (e+ + 2)

f(t) = (e+ + 2))

f(t) = 3 ( + 2))

1

 

3

£x

x              x2

Temukan derivative dari : f(x) =

f'(x )= 2X

f,(x)= x 2e x - ex (2x ) f,(x). x2ex-42xex = xe x (x4 - 2)

x4           x              x

ex (2x)-x2ex       ex (x - 2)

f'(x)= ^^               f'(x) =

4              A V-/- 3

x              x3

4

Temukan derivative dari f(t) = e

f (x) = e3x f (x) = e3x - 3

3x

 

5

6

Temukan derivative dari : f(x) = e

2x2 +1

2x +1

f(x) = e

f '(x) = e2x2+1 (4x)

2

f '(x) = 4xe 2x +1

Temukan derivative dari f (x)= e

f'(x) = e^ -f fa)

dx

f' (x) = e^ • 1(5x) (5)

f' (x)=    5

e

f' (x )= r-2V5x

5e^

V5

x

 

Diferensiasi Logaritma

d In (x) 1

dx x

Jika x = ey. Maka :

d In (x) 1 1 1 dx ~ d (ey) = e7 = x'

dy

 

Aturan Diferensiasi Fungsi Logaritma

Aturan 1:

ln|x| = - (x * 0 )

d

? ix

dx x

Aturan 2:

 

f (x) > 0

 

Contoh

1

2

Temukan derivative dari f(x)= xlnx.

f(x) = xlnx 1

f '(x) = x — + lnx -1 x

f '(x) = 1 + lnx

Temukan derivative dari g(x)= lnx/x

( ) lnx g(x) = -

x

x x- lnx •1            „ , 1 - lnx

g ' (x) = —x-2--g (x) =

2              c? v /     2

x 2          x 2

 

3

4

Temukan derivative dari : y = x2lnx .

(1 ^

y' = x2 1-J + (lnx)(2x)

v x J y' = x + 2xlnx

Atau y' = x(1+2lnx)

Temukan derivative dari y = ln(x + 4)- ln(x - 3)

11           x-3          x+4

y'=—r--r y =

x + 4 x —

3              (x + 4)x — 3) (x + 4)x — 3)

y' = — 7

(x + 4 Xx — 3 )

 

5

Temukan derivative dari : y = ln[(x2 + 1)(x3 + 2)6]

y = ln

(x2 + 1)3 + 2)6 = ln (x2 +1) + ln(x3 + 2)6 = ln(x2 +1)+ 6ln(x3 + 2)

y = ln (x2 +1)+ 6ln (x3 + 2) ' 2x 6 (3x2)

y =(1+(2)

y =

2x (x3 + 2

(x2 +1)x3 + 2)+ (x3 + 2 )2 +1

6 (3x2 )(x2 +1)

y =

2x (x 3 + 2) + 6 (3x2 )(x 2 +1

)

(x2 +1)x3 + 2) (x3 + 2 )x2 + 1)

2x 4 + 4x y = ^--c +

18x 4 + 18x 2

(x2 + 1)x3 + 2) (x3 + 2 )x2 + 1)

'= 20x 4 + 18x 2 + 4x 7 (x2 + 1)x3 + 2)

 

6

Temukan derivative dari : y = x(x + 1)(x2 +1)

Langkah 1 Buat ln pada dua sisi persamaan

ln y = ln x(x + 1)(x2 +1)

Langkah 2 Kembangkan persamaan tersebut

ln y = ln x(x + 1)(x2 +1)

ln y = ln x + ln(x +1) + ln(x2 +1)

Langkah 3 Diferensiasi kedua sisi (eksplisitkan ln y )

ln y = ln x + ln(x +1) + ln(x2 +1)

y' 11 2x

' = — +-+

y x x +1 x2 +1 Langkah 4: Pecahkan y y' = y

(11 2x

A

— +-+ —2-

vx x +1 x2 +1 j

 

Langkah 5: Substitusikan y pada persamaan tersebut.

y = x(x + 1)(x2 +1)

y' = x(x + 1)(x2 +1)

r

1 1 2x

— +-+

V

x x +1 x2 + 1J

y =

x(x + 1)(x2 +1) + x(x + 1)(x2 +1) + 2x[x(x + 1)(x2 +1)

A

y =

x

x +1

x2 +1

J

x (x + 1)(x2 +1) x (x + 1)(x2 +1) + 2xx (x + 1)(x2 +1)]

x + 1

+

A

x2 + 1

J

y' = ((x + 1)x2 + 1) + x(x2 + 1) + 2x [x (x + 1)]) y' = (x3 + x2 + x +1 + x3 + x + 2x3 + 2x2 )

y ' = 4x3 + 3x2 + 2x +1

 

Diferensial Implisit

y = 3 X3 - 4 X +17

y diekspresikan secara explisit sebagai fungsi x.

y3 + xy = 3 x +1

y diekspresikan secara implisit sebagai fungsi x.

[f (x) ]3 + x [X (x) ] = 3 x +1

Diferensial dari fungsi y yang dinyatakan secara implisit disebut diferensial implisit

 

Manfaat Diferensial Implisit

Menemukan slope dari garis tangen dan garis normal

Contoh menemukan slope dari garis tangen dan normal di titik (2,4)

 

Contoh

Temukan diferensial implisit dari

[f (x) ]3 + x [f (x) ] = 3 x +1

3 [f (x)]2 f' (x) + f (x) + xf' (x) = 3

3 y2 y + y + xy' = 3

I

y

+ x ) = 3 - y

' 3 - y

3 y + x

1

 

2

Temukan diferensial implisit dari y2 = x2 + sin(xy)

2y— = 2x + cos(xy) x— + y(1)

dx           V dx j

2 y— = 2 x + cos( xy)(x—) + cos( xy) y dx               dx

2 y — - cos( xy)(x—) = 2 x + cos( xy) y dx               dx

—(2y - x cos(xy)) = 2x + y cos(xy) dx

dy = 2 x + y cos( xy) dx 2y - x cos(xy)

 

3

Temukan diferensial implisit dari x3 + y3 - 9xy = 0

3 x2 + 3 y2 ^ - (9 x^ + y 09) = 0 dx dx

3x2 + 3y2 dy.-9xdy _9y) = 0

dx dx

3y2 ddL-9xddL = 9y-3x2

dx dx

^ (3y2 - 9x) = 9y - 3x2

dx

dy 9 y - 3 x2

dx 3y2 - 9x)

 

4

Temukan diferensial implisit dari y3 + y2 -5y-x2 =-4

d_ dx

d_ dx

y3 + y2 - 5 y - x2 ] = d [-4 ]

y

+

d_ dx

y

d rr -, d

--t\5 y]-

dx

dx

x

= - [-4 ]

dx

3 y2 dy

+ 2 y—-5 ^y - 2 x = 0

dx dx dx

3 y 2 ± + 2 y± - 5 ± = 2 x

dx dx dx

dy dx

(3 y2 + 2 y - 5) = 2 x

dy = 2x dx (3 y + 2 y - 5)

 

Diferensial Parsial

Definisi Derivative Parsial dari Fungsi Dua Variabel Jika z = f(x,y), derivative parsian pertama dari f dinyatakan fx dan fy yaitu :

/ x           f (x + Ax, y)- f (x, y)

X (x v) = lim —_—_

Jx\x>y) ~ Ax Hii0               Ax

, ( \ r f (xy+Ay)-f(xy)

fy(x, y) = Ay 1m 0-—-

 

Definisi Derivative Parsial dari Fungsi Tiga Variabel Jika    w=f(x,y,z), maka derivative parsial

dinyatakansebagai berikut:

dw dx dw dy

dw

~dZ

= fx (x y >z )= Ax liumi 0

= fy (x y,z )= Ay limi 0

f (x + Ax, y, z )-f (x, y, z)

A

= fz (x y, z )= Az Im 0

f (x, y + Ay, z ) -f (x, y, z )

Ay

f (x, y, z + Az) - f (x, y, z)

AA

 

Contoh

Temukan diferensial parsial fx dan fy dari

f (x, y) = 5 x4 - x2 y2 + 2x3 y

Solusi

f (x, y) = 5x* - x2y2 + 2x*y

fx (x, y) = 20x3 - 2y2 x + 6yx2

fy (x, y) = -2 x 2 y + 2 x3

1

 

2

Temukan diferensial parsial fx dan fy dari

f (x,y) = ipad8 titik (2, -2)

x - y

Solusi

f (x, y) = -^at (2, -2)

fx (x y) =

x - y

(x - y)y - xy xy - y2 - xy -y2

(x - y)2 (x - y)2 (x - y)2

f (2 -2)= -(-2^ =n = zi

' (2-(-2))2 16 4

( ) = (x - y)x + xy = x2 - xy + xy

Jy (x' y) = "

x 2

(x - y )2 (x - y )2 (x - y )2

x2 4 1

fy 2-2) = "(x - y)2 ,6 4

 

3

Temukan diferensial parsial fx dan fy dari

f (x, y) = 3xy2 - 2y + 5x2y

Solusi f(x,y) = 3xy2 - 2y + 5x2y

fx (x, y) = 3 y2 +10 xy fxx (x, y) = 10 y

f (x, y) = 3xy2 - 2y + 5x2y fy (x, y) = 6xy - 2 + 5x2

fw(x y) = 6 x

f (x, y) = 3xy2 - 2y + 5x2y fx (x, y) = 3 y2 +10 xy fxy (x, y) = 6 y +10 x

f (x, y) = 3xy2 - 2y + 5x2y fy (x, y) = 6xy - 2 + 5x2 Xvx (x, y) = 6 y +10 x

 

Derivative Tingkat Tinggi

Derivative fungsi f(x) adalah f (x). Jika f '(x) mempunyai derivative, disebut derivative tingakt dua atau f "(x)

Notasif» (x) =

dx

Derivative tingkat 2 mempunyai derivative tingkat tiga dan derivative tingkat tiga mempunyai derivative tingkat empat dst

Notasi f • (x)=f (4)(x)=Zy f

UA          UA          UA

 

Contoh

1

2

Temukan derivative tingkat dua dari

(x -1)(1) - x (1) -1

f ( x ) =

x

x -1

f \ x ) =

f" ( x ) =

(x -1)

(x -1)

d_ dx

(

-1

A

V

(x -1)

J

f' (x) = -2( x -1)-3(1) =

= d ((x --2

(x -1):

Temukan f '''(x) dari : f (x) = 3x5 - 2x3 +14

f'(x) = 15x - 6x

f"(x) = 60x3 -12x

 

3

Temukan derivative tingkat dua dari f (x) =

2              x +1

3              x - 2

2 (3x - 2)-3 (2 x +1) -7      , ,-2

f(x) = -1-\-=-'-^r = -7 (3x-2) 2

(3 x - 2 )2              (3 x - 2 )2 V ^

-3            42

f' (x) = I4 (3 x - 2 )-3 (3 ) = ——,

(3 x - 2 )

4

Temukan f '''(x) dari : f(x) = x2 + 4x + 4

f(x) = 2x + 4 f"(x) = 2

f'' '(x) = 0