BAB 5. TURUNAN - BAB 5 TURUNAN

BAB 5. TURUNAN

Program Studi Teknik Informatika
Fakultas Teknik
Universitas Muhammadiyah Jember

29th April 2018

Ilham Saifudin (TI)

KALKULUS

29th April 2018

1 / 17

Outline

1

Turunan

Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan

Ilham Saifudin (TI)

KALKULUS

29th April 2018

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Turunan

Konsep Turunan

KALKULUS

1

Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan

Ilham Saifudin (TI)

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29th April 2018

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Turunan

Konsep Turunan

Untuk mendefinisikan pengertian garis singgung secara formal, perhatikanlah gambar

di samping kiri. Garis talibusur m1 menghubungkan titik P dan Q1 pada kurva.
Selanjutnya titik Q1 kita gerakkan mendekati titik P. Saat sampai di posisi Q2 ,
talibusurnya berubah menjadi garis m2 . Proses ini diteruskan sampai titik Q1 berimpit
dengan titik P, dan garis talibusurnya menjadi garis singgung m.

Ilham Saifudin (TI)

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29th April 2018

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Turunan

Konsep Turunan

Gradien garis singgung tersebut dapat dinyatakan :
m = lim

h→0

Ilham Saifudin (TI)

f (c + h) − f (c)
= f ′ (c) = y ′
h

KALKULUS

29th April 2018

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Turunan

Definisi turunan

KALKULUS

1

Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan

Ilham Saifudin (TI)

KALKULUS

29th April 2018

6 / 17

Turunan

Definisi turunan

Definisi turunan

Definisi
1

Misalkan f sebuah fungsi real dan x ∈ Df

2

Turunan dari f di titik x, ditulis
f ′ (x) = lim

h→0

f (x + h) − f (x)
h

contoh
Carilah kemiringan garis singgung terhadap y = x 2 − 2x di titik (2, 0)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

7 / 17

Turunan

Definisi turunan

Definisi turunan

Definisi
1

Misalkan f sebuah fungsi real dan x ∈ Df

2

Turunan dari f di titik x, ditulis
f ′ (x) = lim

h→0

f (x + h) − f (x)
h

contoh
Carilah kemiringan garis singgung terhadap y = x 2 − 2x di titik (2, 0)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

7 / 17

Turunan

Definisi turunan

Definisi turunan

Definisi
1

Misalkan f sebuah fungsi real dan x ∈ Df

2

Turunan dari f di titik x, ditulis
f ′ (x) = lim

h→0

f (x + h) − f (x)
h

contoh
Carilah kemiringan garis singgung terhadap y = x 2 − 2x di titik (2, 0)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

7 / 17

Turunan

Definisi turunan

Definisi turunan

Definisi
1

Misalkan f sebuah fungsi real dan x ∈ Df

2

Turunan dari f di titik x, ditulis
f ′ (x) = lim

h→0

f (x + h) − f (x)
h

contoh
Carilah kemiringan garis singgung terhadap y = x 2 − 2x di titik (2, 0)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

7 / 17

Turunan

Definisi turunan

Definisi turunan

Definisi
1

Misalkan f sebuah fungsi real dan x ∈ Df

2

Turunan dari f di titik x, ditulis
f ′ (x) = lim

h→0

f (x + h) − f (x)
h

contoh
Carilah kemiringan garis singgung terhadap y = x 2 − 2x di titik (2, 0)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

7 / 17

Turunan

Aturan turunan

KALKULUS

1

Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan

Ilham Saifudin (TI)

KALKULUS

29th April 2018

8 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan
Aturan turunan
1

Misalkan k sebuah konstanta, maka Dx [k] = 0

2

Dx [x] = 1

3

Dx [x n ] = nx n−1

4

Dx [kf (x)] = kDx [f (x)]

5

Dx [(f ± g)(x)] = Dx [f (x)] ± Dx [g(x)]

6

Dx [(f .g)(x)] = Dx [f (x)].g(x) + f (x).Dx [g(x)]

7

Dx [( gf )(x)] =

Dx [f (x )].g(x )−f (x ).Dx [g(x )]
(g(x )2 )

Aturan turunan fungsi trigonometri
1

Dx [sinx] = cosx , Dx [cosx] = −sinx

2

Dx [tanx] = sec 2 x , Dx [cotx] = −cosec 2 x

3

Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx
Ilham Saifudin (TI)

KALKULUS

29th April 2018

9 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = 5x 2 + sinx, maka f ′ (x) =?

2

Jika f (x) = x 2 .sinx, maka f ′ ( 2 ) =?

3

Jika f (x) =

Q

5x +1
.sinx,
3x −2

Ilham Saifudin (TI)

maka f ′ (1) =?

KALKULUS

29th April 2018

10 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = 5x 2 + sinx, maka f ′ (x) =?

2

Jika f (x) = x 2 .sinx, maka f ′ ( 2 ) =?

3

Jika f (x) =

Q

5x +1
.sinx,
3x −2

Ilham Saifudin (TI)

maka f ′ (1) =?

KALKULUS

29th April 2018

10 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = 5x 2 + sinx, maka f ′ (x) =?

2

Jika f (x) = x 2 .sinx, maka f ′ ( 2 ) =?

3

Jika f (x) =

Q

5x +1
.sinx,
3x −2

Ilham Saifudin (TI)

maka f ′ (1) =?

KALKULUS

29th April 2018

10 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = 5x 2 + sinx, maka f ′ (x) =?

2

Jika f (x) = x 2 .sinx, maka f ′ ( 2 ) =?

3

Jika f (x) =

Q

5x +1
.sinx,
3x −2

Ilham Saifudin (TI)

maka f ′ (1) =?

KALKULUS

29th April 2018

10 / 17

Turunan

Aturan turunan

Aturan turunan

Aturan Rantai
Misalkan y = f (u) dan u = g(x). Jika g terdefinisikan di x dan f terdefinisikan di
u = g(x), maka fungsi komposit f ◦ g, yang didefinisikan oleh (f ◦ g)(x) = f (g(x)),
adalah terdiferensiasikan di x dan (f ◦ g)′ (x) = f ′ (g(x))g ′ (x)

Dx (f (g(x))) =

yakni

f ′ (g(x))g ′ (x)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

11 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = (x 2 − 3x + 5)3 , maka f ′ (x) =?

2

Jika f (x) = sin2 (x 2 − 3x), maka f ′ (x) =?

Ilham Saifudin (TI)

KALKULUS

29th April 2018

12 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = (x 2 − 3x + 5)3 , maka f ′ (x) =?

2

Jika f (x) = sin2 (x 2 − 3x), maka f ′ (x) =?

Ilham Saifudin (TI)

KALKULUS

29th April 2018

12 / 17

Turunan

Aturan turunan

Aturan turunan

Contoh
1

Jika f (x) = (x 2 − 3x + 5)3 , maka f ′ (x) =?

2

Jika f (x) = sin2 (x 2 − 3x), maka f ′ (x) =?

Ilham Saifudin (TI)

KALKULUS

29th April 2018

12 / 17

Turunan

Aturan turunan

Aturan turunan

Turunan tingkat tinggi
Misalkan f (x) sebuah fungsi dan f ′ (x) turunan pertamanya. Turuna kedua dari f
adalah f ”(x) = Dx2 (f ). Dengan cara yang sama turunan ketiga , keempat dst. Salah
satu penggunaan turunan tingkat tinggi adalah pada masalah gerak partikel. Bila S(t)
menyatakan posisi sebuah partikel, maka kecepatannya adalah v (t) = S ′ (t) dan
percepatannya a(t) = v ′ (t) = S”(t)

Ilham Saifudin (TI)

KALKULUS

29th April 2018

13 / 17

Turunan

Aplikasi turunan

KALKULUS

1

Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan

Ilham Saifudin (TI)

KALKULUS

29th April 2018

14 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f’(x)
1

Gradien g singgung : m = y ′

2

fungsi naik : y ′ > 0

3

fungsi turun : y ′ < 0

4

fungsi stasioner : y ′ = 0

5

kecepatan : v ′ =

6

percepatan : a′ =

Ilham Saifudin (TI)

ds
dt
dv
dt

= S′
= v ′ = S”

KALKULUS

29th April 2018

15 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f”(x)
Uji jenis
1

maximum : y” > 0

2

minimum : y” < 0

3

titik belok : y” = 0

Ilham Saifudin (TI)

KALKULUS

29th April 2018

16 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f”(x)
Uji jenis
1

maximum : y” > 0

2

minimum : y” < 0

3

titik belok : y” = 0

Ilham Saifudin (TI)

KALKULUS

29th April 2018

16 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f”(x)
Uji jenis
1

maximum : y” > 0

2

minimum : y” < 0

3

titik belok : y” = 0

Ilham Saifudin (TI)

KALKULUS

29th April 2018

16 / 17

Turunan

Aplikasi turunan

Aplikasi turunan

y=f”(x)
Uji jenis
1

maximum : y” > 0

2

minimum : y” < 0

3

titik belok : y” = 0

Ilham Saifudin (TI)

KALKULUS

29th April 2018

16 / 17

Turunan

Aplikasi turunan

Thank You

Ilham Saifudin (TI)

KALKULUS

29th April 2018

17 / 17