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BAB 5. TURUNAN
Program Studi Teknik Informatika
Fakultas TeknikUniversitas Muhammadiyah Jember
2nd May 2017
Ilham Saifudin (TI) KALKULUS 2nd May 2017 1 / 17
Outline
1 Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan
Ilham Saifudin (TI) KALKULUS 2nd May 2017 2 / 17
Turunan Konsep Turunan
KALKULUS
1 Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan
Ilham Saifudin (TI) KALKULUS 2nd May 2017 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 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) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 5 / 17
Turunan Definisi turunan
KALKULUS
1 Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan
Ilham Saifudin (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 7 / 17
Turunan Aturan turunan
KALKULUS
1 Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan
Ilham Saifudin (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 13 / 17
Turunan Aplikasi turunan
KALKULUS
1 Turunan
Konsep Turunan
Definisi turunan
Aturan turunan
Aplikasi turunan
Ilham Saifudin (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 (TI) KALKULUS 2nd May 2017 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 2nd May 2017 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 2nd May 2017 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 2nd May 2017 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 2nd May 2017 16 / 17
Turunan Aplikasi turunan
Thank You
Ilham Saifudin (TI) KALKULUS 2nd May 2017 17 / 17
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