BAB 6. TURUNAN

Jurusan Manajemen Informatika

Fakultas TeknikUniversitas Muhammadiyah Jember

21st November 2016

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 1 / 17

Outline

1 Turunan

Konsep Turunan

Definisi turunan

Aturan turunan

Aplikasi turunan

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 2 / 17

Turunan Konsep Turunan

MATEMATIKA DASAR

1 Turunan

Konsep Turunan

Definisi turunan

Aturan turunan

Aplikasi turunan

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 3 / 17

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 titikP. 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 (MI) MATEMATIKA DASAR 21st November 2016 4 / 17

Turunan Konsep Turunan

Gradien garis singgung tersebut dapat dinyatakan :

m = limh→0

f (c + h) − f (c)

h= f ′(c) = y ′

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 5 / 17

Turunan Definisi turunan

MATEMATIKA DASAR

1 Turunan

Konsep Turunan

Definisi turunan

Aturan turunan

Aplikasi turunan

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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) = limh→0

f (x + h) − f (x)

h

contoh

Carilah kemiringan garis singgung terhadap y = x2− 2x di titik (2, 0)

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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) = limh→0

f (x + h) − f (x)

h

contoh

Carilah kemiringan garis singgung terhadap y = x2− 2x di titik (2, 0)

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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) = limh→0

f (x + h) − f (x)

h

contoh

Carilah kemiringan garis singgung terhadap y = x2− 2x di titik (2, 0)

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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) = limh→0

f (x + h) − f (x)

h

contoh

Carilah kemiringan garis singgung terhadap y = x2− 2x di titik (2, 0)

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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) = limh→0

f (x + h) − f (x)

h

contoh

Carilah kemiringan garis singgung terhadap y = x2− 2x di titik (2, 0)

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 7 / 17

Turunan Aturan turunan

MATEMATIKA DASAR

1 Turunan

Konsep Turunan

Definisi turunan

Aturan turunan

Aplikasi turunan

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 8 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Aturan turunan

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

2 Dx [x] = 1

3 Dx [xn] = nxn−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 [(fg )(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] = sec2x , Dx [cotx] = −cosec2x

3 Dx [secx] = secxtanx , Dx [cosecx] = −cosecxcotx

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 9 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = 5x2 + sinx , maka f ′(x) =?

2 Jika f (x) = x2.sinx , maka f ′(

Q2 ) =?

3 Jika f (x) = 5x+13x−2 .sinx , maka f ′(1) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 10 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = 5x2 + sinx , maka f ′(x) =?

2 Jika f (x) = x2.sinx , maka f ′(

Q2 ) =?

3 Jika f (x) = 5x+13x−2 .sinx , maka f ′(1) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 10 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = 5x2 + sinx , maka f ′(x) =?

2 Jika f (x) = x2.sinx , maka f ′(

Q2 ) =?

3 Jika f (x) = 5x+13x−2 .sinx , maka f ′(1) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 10 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = 5x2 + sinx , maka f ′(x) =?

2 Jika f (x) = x2.sinx , maka f ′(

Q2 ) =?

3 Jika f (x) = 5x+13x−2 .sinx , maka f ′(1) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 10 / 17

Turunan Aturan turunan

Aturan turunan

Aturan RantaiMisalkan 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) yakniDx(f (g(x))) = f ′(g(x))g′(x)

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 11 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = (x2− 3x + 5)3, maka f ′(x) =?

2 Jika f (x) = sin2(x2− 3x), maka f ′(x) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 12 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = (x2− 3x + 5)3, maka f ′(x) =?

2 Jika f (x) = sin2(x2− 3x), maka f ′(x) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 12 / 17

Turunan Aturan turunan

Aturan turunan

Contoh

1 Jika f (x) = (x2− 3x + 5)3, maka f ′(x) =?

2 Jika f (x) = sin2(x2− 3x), maka f ′(x) =?

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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) = D2x (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 (MI) MATEMATIKA DASAR 21st November 2016 13 / 17

Turunan Aplikasi turunan

MATEMATIKA DASAR

1 Turunan

Konsep Turunan

Definisi turunan

Aturan turunan

Aplikasi turunan

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 ′ = dsdt = S′

6 percepatan : a′ = dvdt = v ′ = S”

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 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 (MI) MATEMATIKA DASAR 21st November 2016 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 (MI) MATEMATIKA DASAR 21st November 2016 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 (MI) MATEMATIKA DASAR 21st November 2016 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 (MI) MATEMATIKA DASAR 21st November 2016 16 / 17

Turunan Aplikasi turunan

Thank You

Ilham Saifudin (MI) MATEMATIKA DASAR 21st November 2016 17 / 17