Pada pertemuan sebelumnya, kita telah membahas awal dari turunan fungsi dan hubungannya dengan limit. Jika kalian sudah lupa, silakan klik pada link berikut ini https://bdr.masbied.com/bdr-genap-kelas-xi/2266/xi-7-1-turunan-fungsi-aljabar/
Di pembahasan itu, kita menemukan turunan fungsi berikut ini…
- Turunan dari $f(x)$ adalah $f'(x) = \displaystyle \lim_{\Delta x \to 0}\frac{f(x_1 + \Delta x)-f(x)}{\Delta x}$
- Jika $f(x) = C \to f'(x) = 0$, dengan $C$ adalah konstanta.
- Jika $f(x) = \textcolor{red}{a}x^\textcolor{red}{1} \to f'(x) = a = 1.\textcolor{red}{a}.x^\textcolor{red}{1-1} $
- Jika $f(x) = \textcolor{red}{a}x^\textcolor{red}{2} \to f'(x)= 2ax = 2.\textcolor{red}{a}.x^\textcolor{red}{2-1} $
- Jika $f(x)= \textcolor{red}{a}x^\textcolor{red}{3} \to f'(x) = 3ax^2 = 3.\textcolor{red}{a}.x^\textcolor{red}{3-1} $
Berdasarkan kesimpulan nomor 2 s.d 5, bagaimanakah turunan dari $f(x)=ax^n$?
Jika melihat polanya, maka kita dapat rumus umum turunan berikut
$\boxed{f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}}$
Contoh 1 :
Turunan pertama dari $f(x) = 1x^{1000}$ adalah ….
Jawab :
$f(x)=\textcolor{red}{a}x^\textcolor{red}{n} \to f'(x) = n.\textcolor{red}{a}.x^\textcolor{red}{n-1}$
$\begin{aligned}f(x) = x^{1000} \to f'(x) &= 1000.x^{(1000-1)} \\ &= 1000 x^{999} \end{aligned}$
Bener nggak nih?
Coba kita buktikan dengan konsep limit!
Ingat, $f'(x) =\displaystyle \lim_{\Delta x \to 0}\frac{f(x + \Delta x)-\textcolor{blue}{f(x)}}{\Delta x}$
Karena
$\textcolor{blue}{f(x) = \textcolor{red}{x}^{1000}}$
$\begin{aligned}\textcolor{red}{f(x+\Delta x)} &= {(\textcolor{red}{x+\Delta x})}^{1000} \end{aligned}$
Wah, berapa tuh ${(\textcolor{red}{x+\Delta x})}^{1000}$?
Dengan menggunakan segitiga pascal, bisa kamu temukan bahwa …
$\begin{aligned}(a+b)^0&=1 \\ (a+b)^1 &= 1a + 1b \\ (a+b)^2 &=1a^2+2ab+1b^2 \\ (a+b)^3 &= 1a^3+3a^2b+3ab^2+1b^3 \\ \vdots \\ (a+b)^{1000}&=1a^{1000}+1000a^{999}b+\cdots+1b^{1000} \end{aligned}$
Sehingga bisa dilihat
$\textcolor{red}{f(x+\Delta x)={(x+\Delta x)}^{1000} = x^{1000}+1000.x^{999}.\Delta x+\cdots+ (\Delta x)^{1000}}$
Ingat, $\Delta x$ pada turunan mendekati $0$, sehingga pada saat mengerjakan limit nanti, nilainya juga akan mendekati $0$.
Bisa kita tulis menjadi
$\begin{aligned}f'(x)&=\displaystyle \lim_{\Delta x \to 0}\frac{\textcolor{red}{f(x + \Delta x)}-\textcolor{blue}{f(x)}}{\Delta x} \\ &= \lim_{\Delta x \to 0}\frac{\textcolor{red}{(x+\Delta x)^{1000}}-x^{1000}}{\Delta x} \\ &=\lim_{\Delta x \to 0}\frac{\textcolor{red}{x^{1000}+1000.x^{999}.\Delta x+\cdots+ (\Delta x)^{1000}} – x^{1000}}{\Delta x} \\ &=\lim_{\Delta x \to 0}\frac{\textcolor{red}{x^{1000}}+1000.x^{999}.\Delta x+\cdots+ (\Delta x)^{1000} – \textcolor{red}{x^{1000}}}{\Delta x}\\ &=\lim_{\Delta x \to 0}\frac{1000.x^{999}.\Delta x + \cdots +(\Delta x)^{999}}{\Delta x} \\ &= \lim_{\Delta x \to 0}\frac{\textcolor{red}{\Delta x}(1000.x^{999} + \cdots +(\Delta x)^{999})}{\textcolor{red}{\Delta x}} \\ &= \lim_{\Delta x \to 0}{(1000.x^{999}+\cdots+(\Delta x)^{999})} \to \textcolor{blue}{{\Delta x}^{999}=0} \\ &= 1000x^{999}\end{aligned}$
Oke …
Jadi, Jika $f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}$
Contoh 2 :
Turunan dari fungsi $g(x) = 2x^{99}$ adalah …
Jawab :
$f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}$
$g(x) = \textcolor{red}{2}x^\textcolor{blue}{{99}} \to g'(x)= \textcolor{blue}{99}.\textcolor{red}{2}.x^\textcolor{blue}{99-1} = 198x^{98}$
Contoh 3 :
Turunan dari fungsi $f(x) = 10x^4$ adalah …
Jawab :
$f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}$
$f(x)=\textcolor{red}{10}x^\textcolor{blue}{4} \to f'(x) = \textcolor{blue}{4}.\textcolor{red}{10}.x^\textcolor{blue}{4-1} = 40x^3$
Contoh 4 :
Tentukan turunan dari $f(x) = 5x^{\frac{1}{2}}$!
Jawab :
$f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}$
$f(x)=\textcolor{red}{5}x^\textcolor{blue}{\frac{1}{2}} \to f'(x) = \textcolor{blue}{\frac{1}{2}}.\textcolor{red}{5}.x^\textcolor{blue}{{\frac{1}{2}}-1} = \frac{5}{2}x^{\frac{-1}{2}}$
Jadi, ingat selalu bahwa $\boxed{f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}}$
Sip.
Kita lanjut ke materi berikutnya!
B. Rumus-Rumus Turunan Fungsi Aljabar
1. Turunan konstanta $f(x)=C$
$\boxed{f(x) = C \to f'(x) = 0}$
Contoh 5 :
Tentukan turunan dari :
a. $f(x) = 5$
b. $f(x) = 100$
c. $f(x) = -5$
d. $f(x) = \frac{1}{2}$
Jawab :
a. $f(x) = 5 \to f'(x)=0$
b. $f(x) = 100 \to f'(x)=0$
c. $f(x) = -5 \to f'(x)=0 $
d. $f(x) = \frac{1}{2} \to f'(x)=0$
2. Turunan $f(x)=ax$
$\boxed{f(x)=ax \to f'(x)=a}$
Contoh 5 :
Tentukan turunan dari
a. $f(x) = 5x$
b. $f(x) = -2x$
c. $f(x) = 21x$
Jawab :
a. $f(x) = 5x \to f'(x) = 5$
b. $f(x) = -2x \to f'(x) = -2$
c. $f(x) = 21x \ to f'(x) = 21$
3. Turunan $f(x)=ax^n$
$\boxed{f(x)=\textcolor{red}{a}x^\textcolor{blue}{n} \to f'(x) = \textcolor{blue}{n}.\textcolor{red}{a}.x^\textcolor{blue}{n-1}}$
Contoh 6 :
Tentukan turunan dari fungsi berikut!
a. $f(x) = 5x^3$
b. $f(x) = 12x^{-2}$
c. $f(x) = -4x^7$
d. $f(x) = -2x^{-3}$
Jawab :
a. $f(x) = 5x^3 \to f'(x) = 3.5.x^{3-1} = 15x^2$
b. $f(x) = 12x^{-2} \to f'(x) = (-2).12.x^{-2-1} = -24x^{-3}$
c. $f(x) = -4x^7 \to f'(x) = 7.(-4).x^{7-1} = -28x^6$
d. $f(x) = -2x^{-3} \to f'(x) = (-3).(-2).x^{-3-1} = 6x^{-4}$
4. Turunan Penjumlahan $f(x) = u(x) + v(x)$
$\boxed{f(x) = u(x)+ v(x) \to f'(x) =u'(x) + v'(x)}$
Contoh 7 :
Tentukan turunan dari :
a. $f(x) = 5x^2 + 7x$
b. $f(x) = 12x^3 + 6x^2 + 3x$
c. $f(x) = -3x^2 + 4x$
d. $f(x) = -4x^-3 + 2x^2$
Jawab :
a. $f(x) = 5x^2 + 7x$
Misalkan :
$\begin{aligned}u(x) &= 5x^2 \to u'(x) = 2.5.x^{2-1} = 10x \\ v(x) &=7x \to v'(x) = 7 \end{aligned}$
$f(x) = 5x^2 + 7x$
$\begin{aligned}f'(x) &= u'(x) + v'(x) \\ &= 10x + 7 \end{aligned}$
b. $f(x) = 12x^3 + 6x^2 + 3x$
Misalkan :
$\begin{aligned}u(x) &= 12x^3 \to u'(x) = 3.12.x^{3-1} = 36x^2 \\ v(x) &=6x^2 \to v'(x) = 2.6.x^{2-1} = 12x \\ w(x) &= 3x \to w'(x) = 3 \end{aligned}$
$f(x) = 12x^3 + 6x^2 + 3x$
$\begin{aligned}f'(x) &= u'(x) + v'(x) + w'(x) \\ &= 36x^2 + 12x + 3 \end{aligned}$
c. $f(x) = -3x^2 + 4x$
Langsung saja yah …
$f'(x) = 2.(-3).x^{2-1} + 4 = -6x + 4$
d. $f(x) = -4x^-3 + 2x^2$
Langsung juga, ditulis
$f'(x) = (-3).(-4).x^{-3-1} + 2.2x^{2-1} = 12x^{-4} + 4x$
5. Turunan Pengurangan $f(x) = u(x) – v(x)$
Jika $\boxed{f(x) = u(x)- v(x) \to f'(x) =u'(x) – v'(x)}$
Tentukan turunan dari :
a. $f(x) = 5x^2 – 7x $
b. $f(x) = 12x^3 – 6x^2 – 3x$
c. $f(x) = -3x^2 – 4x$
d. $f(x) = -4x^-3 – 2x^2$
Jawab :
a. $f(x) = 5x^2 – 7x$
$\begin{aligned}f'(x) &= 2.5.x^{2-1} – 7 \\ &= 10x – 7\end{aligned}$
b. $f(x) = 12x^3 – 6x^2 – 3x$
$\begin{aligned}f'(x) &= 3.12.x^{3-1} – 2.6.x^{2-1} -3 \\ &= 36x^2 – 12x – 3\end{aligned}$
c. $f(x) = -3x^2 – 4x$
$\begin{aligned}f'(x) &= 2.(-3).x^{2-1} – 4 \\ &= -6x – 4\end{aligned}$
d. $f(x) = -4x^-3 – 2x^2$
$\begin{aligned}f'(x) &= (-3).(-4).x^{-3-1} – 2.2.x^{2-1} \\ &= 12x^{-4} – 4x\end{aligned}$
Siiplah ..
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