Turunan Fungsi Aljabar

Rumus Umum Turunan Fungsi

Aturan umum turunan fungsi $f\left( x \right)$ dapat didefinisikan sebagai berikut.

Definisi

Misalkan diketahui fungsi $y = f\left( x \right)$ yang terdefinisi pada daerah asal ${D_f} = \left\{ {x|x \in \Re } \right\}$. Turunan fungsi $f\left( x \right)$ terhadap $x$ ditentukan oleh:

\[\boxed{f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}}\]

dengan catatan jika nilai limit itu ada.

Ungkapan matematika $f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$ dikenal sebagai rumus umum turunan fungsi $f\left( x \right)$.

Contoh

Carilah turunan atau $f'\left( x \right)$ untuk fungsi-fungsi berikut:

1. $f\left( x \right) = 2x + 1$
2. $f\left( x \right) = 2\sqrt x ,x \ge 0$
3. $f\left( x \right) = {x^2} + x$
4. $f\left( x \right) = \frac{3}{{x + 2}}$

Jawab

Dengan menggunakan rumus umum turunan $f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$ diperoleh

1. Untuk fungsi $f\left( x \right) = 2x + 1$
$\begin{array}{l} f'\left( x \right) &= \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {2\left( {x + h} \right) + 1} \right] - \left( {2x + 1} \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{2x + 2h + 1 - 2x - 1}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{2h}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} 2\\ &= 2 \end{array}$

Jadi, turunan fungsi $f\left( x \right) = 2x + 1$ adalah $f'\left( x \right) = 2$.

2. Untuk fungsi $f\left( x \right) = 2\sqrt x ,x \ge 0$
$\begin{array}{l} f'\left( x \right) &= \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\sqrt {x + h} - \sqrt x }}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\sqrt {x + h} - \sqrt x }}{h} \times \frac{{\sqrt {x + h} + \sqrt x }}{{\sqrt {x + h} + \sqrt x }}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\left( {x + h} \right) - x}}{{h\left( {\sqrt {x + h} + \sqrt x } \right)}}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\cancel{h}}}{{\cancel{h}\left( {\sqrt {x + h} + \sqrt x } \right)}}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{1}{{\left( {\sqrt {x + h} + \sqrt x } \right)}}\\ &= \frac{1}{{\sqrt {x + 0} + \sqrt x }}\\ &= \frac{1}{{2\sqrt x }} \end{array}$

Jadi, turunan fungsi $f\left( x \right) = 2\sqrt x ,x \ge 0$ adalah $f'\left( x \right) = \frac{1}{{2\sqrt x }}$.

3. Untuk fungsi $f\left( x \right) = {x^2} + x$
$\begin{array}{l} f'\left( x \right) &= \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {{{\left( {x + h} \right)}^2} + \left( {x + h} \right)} \right] - \left( {{x^2} + x} \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\left[ {{x^2} + 2xh + {h^2} + x + h} \right] - \left( {{x^2} + x} \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{{x^2} + 2xh + {h^2} + x + h - {x^2} - x}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{2xh + {h^2} + h}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} 2x + h + 1\\ &= 2x + 0 + 1\\ &= 2x + 1 \end{array}$

Jadi, turunan fungsi $f\left( x \right) = {x^2} + x$ adalah $f'\left( x \right) = 2x + 1$.

4. Untuk fungsi $f\left( x \right) = \frac{3}{{x + 2}}$
$\begin{array}{l} f'\left( x \right) &= \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\frac{3}{{\left( {x + h} \right) + 2}} - \frac{3}{{x + 2}}}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\frac{{3\left( {x + 2} \right) - 3\left[ {\left( {x + h} \right) + 2} \right]}}{{\left[ {\left( {x + h} \right) + 2} \right]\left( {x + 2} \right)}}}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{\frac{{ - 3h}}{{\left[ {\left( {x + h} \right) + 2} \right]\left( {x + 2} \right)}}}}{h}\\ &= \mathop {\lim }\limits_{h \to 0} \frac{{ - 3}}{{\left[ {\left( {x + h} \right) + 2} \right]\left( {x + 2} \right)}}\\ &= \frac{{ - 3}}{{\left[ {\left( {x + 0} \right) + 2} \right]\left( {x + 2} \right)}}\\ &= \frac{{ - 3}}{{{{\left( {x + 2} \right)}^2}}} \end{array}$

Jadi, turunan fungsi $f\left( x \right) = \frac{3}{{x + 2}}$ adalah $f'\left( x \right) = \frac{{ - 3}}{{{{\left( {x + 2} \right)}^2}}}$.

Rumus-Rumus Turunan Fungsi Aljabar

1. Jika $f\left( x \right) = k$ dengan $k$ adalah konstanta real, maka turunan $f\left( x \right)$ adalah \[f'\left( x \right) = 0\].
2. Jika $f\left( x \right)$ adalah fungsi identitas atau $f\left( x \right) = x$ maka \[f'\left( x \right) = 1\].
3. Jika $f\left( x \right) = a{x^n}$ dengan $a$ konstanta real tidak nol dan $n$ bilangan real, maka \[f'\left( x \right) = an{x^{n - 1}}\].
4. Jika $f\left( x \right) = ku\left( x \right)$ dengan $k$ konstanta real dan $u\left( x \right)$ fungsi dari $x$ yang mempunyai turunan $u'\left( x \right)$, maka \[f'\left( x \right) = ku'\left( x \right)\].
5. Jika $f\left( x \right) = u\left( x \right) \pm v\left( x \right)$, dengan $u\left( x \right)$ dan $v\left( x \right)$ masing-masing mempunyai turunan $u'\left( x \right)$ dan $v'\left( x \right)$, maka \[f'\left( x \right) = u'\left( x \right) \pm v'\left( x \right)\].
6. Jika $f\left( x \right) = u\left( x \right).v\left( x \right)$, dengan $u\left( x \right)$ dan $v\left( x \right)$ masing-masing mempunyai turunan $u'\left( x \right)$ dan $v'\left( x \right)$, maka
   \[f'\left( x \right) = u'\left( x \right).v\left( x \right) + v'\left( x \right).u\left( x \right)\]
7. Jika $f\left( x \right) = \frac{{u\left( x \right)}}{{v\left( x \right)}}$, dengan $u\left( x \right)$ dan $v\left( x \right)$ masing-masing mempunyai turunan $u'\left( x \right)$ dan $v'\left( x \right)$, maka
   \[f'\left( x \right) = \frac{{u'\left( x \right).v\left( x \right) - v'\left( x \right).u\left( x \right)}}{{{{\left[ {v\left( x \right)} \right]}^2}}}\]
8. Jika $f\left( x \right) = {\left[ {u\left( x \right)} \right]^n}$, dengan ${u\left( x \right)}$ adalah fungsi dari $x$ yang mempunyai turunan $u'\left( x \right)$ dan $n$ bilangan real, maka \[f'\left( x \right) = n{\left[ {u\left( x \right)} \right]^{n - 1}}.u'\left( x \right)\].

Contoh

Carilah turunan atau $f'\left( x \right)$ untuk fungsi-fungsi berikut:

1. $f\left( x \right) = 3{x^3} + 2{x^2} - 5x + 4$
2. $f\left( x \right) = \left( {{x^2} + x} \right)\left( {{x^3} - 1} \right)$
3. $f\left( x \right) = \frac{{3x + 2}}{{{x^2} - 1}}$
4. $f\left( x \right) = \sqrt x \left( {x\sqrt x - 1} \right)$
5. $f\left( x \right) = {\left( {{x^2} - 4x + 5} \right)^6}$

Jawab

1. $f\left( x \right) = 3{x^3} + 2{x^2} - 5x + 4$
$\begin{array}{l} f'\left( x \right) &= 3.3.{x^{3 - 1}} + 2.2.{x^{2 - 1}} - 5\\ &= 9{x^2} + 4x - 5 \end{array}$

Jadi, turunan fungsi $f\left( x \right) = 3{x^3} + 2{x^2} - 5x + 4$ adalah $f'\left( x \right) = 9{x^2} + 4x - 5$.

2. $f\left( x \right) = \left( {{x^2} + x} \right)\left( {{x^3} - 1} \right)$
$\begin{array}{l} u\left( x \right) &= {x^2} + x \Rightarrow u'\left( x \right) = 2x + 1\\ v\left( x \right) &= {x^3} - 1 \Rightarrow v'\left( x \right) = 3{x^2}\\ f'\left( x \right) &= u'\left( x \right).v\left( x \right) + v'\left( x \right).u\left( x \right)\\ &= \left( {2x + 1} \right).3{x^2} + 3{x^2}.\left( {{x^2} + x} \right)\\ &= 6{x^3} + 3{x^2} + 3{x^4} + 3{x^3}\\ &= 3{x^4} + 9{x^3} + 3{x^2} \end{array}$

Jadi, turunan fungsi $f\left( x \right) = \left( {{x^2} + x} \right)\left( {{x^3} - 1} \right)$ adalah $f'\left( x \right) = 3{x^4} + 9{x^3} + 3{x^2}$.

3. $f\left( x \right) = \frac{{3x + 2}}{{{x^2} - 1}}$
$\begin{array}{l} u\left( x \right) &= 3x + 2 \Rightarrow u'\left( x \right) = 3\\ v\left( x \right) &= {x^2} - 1 \Rightarrow v'\left( x \right) = 2x\\ f'\left( x \right) &= \frac{{u'\left( x \right).v\left( x \right) - v'\left( x \right).u\left( x \right)}}{{{{\left[ {v\left( x \right)} \right]}^2}}}\\ &= \frac{{3.\left( {{x^2} - 1} \right) - 2x.\left( {3x + 2} \right)}}{{{{\left( {{x^2} - 1} \right)}^2}}}\\ &= \frac{{3{x^2} - 3 - 6{x^2} - 4x}}{{{{\left( {{x^2} - 1} \right)}^2}}}\\ &= \frac{{ - 3{x^2} - 4x - 3}}{{{{\left( {{x^2} - 1} \right)}^2}}} \end{array}$

Jadi, turunan fungsi $f\left( x \right) = \frac{{3x + 2}}{{{x^2} - 1}}$ adalah $f'\left( x \right) = \frac{{ - 3{x^2} - 4x - 3}}{{{{\left( {{x^2} - 1} \right)}^2}}}$.

4. $f\left( x \right) = \sqrt x \left( {x\sqrt x - 1} \right)$
$\begin{array}{l} u\left( x \right) &= \sqrt x \Rightarrow u'\left( x \right) = \frac{1}{{2\sqrt x }}\\ v\left( x \right) &= x\sqrt x - 1 \Rightarrow v'\left( x \right) = \frac{3}{2}\sqrt x \\ f'\left( x \right) &= u'\left( x \right).v\left( x \right) + v'\left( x \right).u\left( x \right)\\ &= \frac{1}{{2\sqrt x }}.\left( {x\sqrt x - 1} \right) + \frac{3}{2}\sqrt x .\sqrt x \\ &= \frac{1}{2}x - \frac{1}{{2\sqrt x }} + \frac{3}{2}x\\ &= 2x - \frac{1}{{2\sqrt x }} \end{array}$

Jadi, turunan fungsi $f\left( x \right) = \sqrt x \left( {x\sqrt x - 1} \right)$ adalah $f'\left( x \right) = 2x - \frac{1}{{2\sqrt x }}$.

5. $f\left( x \right) = {\left( {{x^2} - 4x + 5} \right)^6}$
$\begin{array}{l} f'\left( x \right) &= 6.\left( {2x - 4} \right).{\left( {{x^2} - 4x + 5} \right)^{6 - 1}}\\ &= \left( {12x - 24} \right){\left( {{x^2} - 4x + 5} \right)^5} \end{array}$

Jadi, turunan fungsi $f\left( x \right) = {\left( {{x^2} - 4x + 5} \right)^6}$ adalah $f'\left( x \right) = \left( {12x - 24} \right){\left( {{x^2} - 4x + 5} \right)^5}$.

Turunan Ke-n dari Suatu Fungsi

Notasi-notasi untuk turunan pertama, turunan kedua, turunan ketiga, sampai turunan ke-n dari fungsi $y = f\left( x \right)$ disajikan pada tabel berikut:

Jenis Turunan	Notasi yang Digunakan
Turunan Pertama	$y'$ atau $f'\left( x \right)$ atau $\frac{{dy}}{{dx}}$ atau $\frac{{df}}{{dx}}$
Turunan Kedua	$y''$ atau $f''\left( x \right)$ atau $\frac{{{d^2}y}}{{d{x^2}}}$ atau $\frac{{{d^2}f}}{{d{x^2}}}$
Turunan Ketiga	$y'''$ atau $f'''\left( x \right)$ atau $\frac{{{d^3}f}}{{d{x^3}}}$ atau $\frac{{{d^3}f}}{{d{x^3}}}$
...	...
Turunan Ke-n	${y^{\left( n \right)}}$ atau ${f^{\left( n \right)}}\left( x \right)$ atau $\frac{{{d^n}y}}{{d{x^n}}}$ atau $\frac{{{d^n}f}}{{d{x^n}}}$

Contoh

Carilah turunan pertama, turunan kedua, dan turunan ketiga dari fungsi $f\left( x \right) = {\left( {2x - 3} \right)^9}$

Jawab

• Turunan pertama
$\begin{array}{l} f'\left( x \right) &= 9.2.{\left( {2x - 3} \right)^{9 - 1}}\\ &= 18{\left( {2x - 3} \right)^8} \end{array}$
• Turunan kedua
$\begin{array}{l} f''\left( x \right) &= 18.8.2.{\left( {2x - 3} \right)^{8 - 1}}\\ &= 288{\left( {2x - 3} \right)^7} \end{array}$
• Turunan ketiga
$\begin{array}{l} f'''\left( x \right) &= 288.7.2.{\left( {2x - 3} \right)^{7 - 1}}\\ &= 4.032{\left( {2x - 3} \right)^6} \end{array}$