Tugas Media dan Teknologi Pembelajaran Dengan Menggunakan Firemath Materi Fungsi Turunan

FUNGSI TURUNAN

Misalkan $\text{f}\left(x\right)$ suatu fungsi dengan daerah asal terdefinisi D yang mengandung interval terbuka yang memuat $x$ maka turunan pertama dari f ditulis ${\text{f}}^{\prime }\left(x\right)$ didefinisikan:

$${\text{f}}^{\prime }\left(x\right)=\underset{\text{h}\to 0}{\text{Lim}}\frac{\text{f}\left(x+\text{h}\right)-\text{f}\left(x\right)}{\text{h}}$$

A. Teorema Turunan

1.   $\text{f}\left(x\right)=x^n\mathrm{ maka }{\text{f}}^{\prime }\left(x\right)=n\text{.}{x}^{n-1}$       2.  $\text{f}\left(x\right)=a\text{.}{x}^n\mathrm{ maka }{\text{f}}^{\prime }\left(x\right)=n\text{.}{a\text{.}x}^{n-1}$       3.  ${\left[\text{f}\left(x\right)\pm \text{g}\left(x\right)\right]}^{\prime }={\text{f}}^{\prime }\left(x\right)\pm {\text{g}}^{\prime }\left(x\right)$       4.  $c\text{.}\text{f}\left(x\right)=c\text{.}{\text{f}}^{\prime }\left(x\right)$, untuk c suatu bilangan       5.  ${\left[\text{f}\left(x\right)\text{.}\text{g}\left(x\right)\right]}^{\prime }={\text{f}}^{\prime }\left(x\right)\text{.}{\text{g}\left(x\right)+\text{f}\left(x\right)\text{.}\text{g}}^{\prime }\left(x\right)$       6.  $\left[\frac{\text{f}\left(x\right)}{\text{g}\left(x\right)}\right]=\frac{{\text{f}}^{\prime }\left(x\right)\text{.}\text{g}\left(x\right)-\text{f}\left(x\right)\text{.}{\text{g}}^{\prime }\left(x\right)}{{\left[\text{g}\left(x\right)\right]}^2}$

Contoh:
Tentukan turunan pertama dari:
1. $\text{f}\left(x\right)=2x^3-4x^2+3x+5$

2. $\text{f}\left(x\right)=\left(2x^3-x\right)\left(x^4+3x\right)$

Penyelesaian no. 1:
$\text{f}\left(x\right)=2x^3-4x^2+3x+5$
${\text{f}}^{\prime }\left(x\right)=2\left(3x^{3-1}\right)-4\left(2x^{2-1}\right)+3\left(x^{1-1}\right)+0$
${\text{f}}^{\prime }\left(x\right)=2\left(3x^2\right)-4\left(2x\right)+3\left(1\right)+0$
${\text{f}}^{\prime }\left(x\right)=6x^2-8x+3$

Penyelesaian no. 2:
Cara I
$\text{f}\left(x\right)=\left(2x^3-x\right)\left(x^4+3x\right)\to dikalikan\; terlebih\; dahulu$
$\text{f}\left(x\right)=2x^7-x^5+6x^4-3x^2$
${\text{f}}^{\prime }\left(x\right)=2\left(7x^{7-1}\right)-\left(5x^{5-1}\right)+6\left(4x^{4-1}\right)-3\left(2x^{2-1}\right)$
${\text{f}}^{\prime }\left(x\right)=2\left(7x^6\right)-\left(5x^4\right)+6\left(4x^3\right)-3\left(2x\right)$
${\text{f}}^{\prime }\left(x\right)=14x^6-5x^4+24x^3-6x$

Cara II
Dengan menggunakan teorema ke-5, yaitu  ${\left[\text{f}\left(x\right)\text{.}\text{g}\left(x\right)\right]}^{\prime }={\text{f}}^{\prime }\left(x\right)\text{.}{\text{g}\left(x\right)+\text{f}\left(x\right)\text{.}\text{g}}^{\prime }\left(x\right)$
$\text{f}\left(x\right)=2x^3-x$
${\text{f}}^{\prime }\left(x\right)=2\left(3x^2\right)-1$
${\text{f}}^{\prime }\left(x\right)=6x^2-1$

$\text{g}\left(x\right)=x^4+3x$
${\text{g}}^{\prime }\left(x\right)=4x^3+3$

Sehingga
${\left[\text{f}\left(x\right)\text{.}\text{g}\left(x\right)\right]}^{\prime }={\text{f}}^{\prime }\left(x\right)\text{.}{\text{g}\left(x\right)+\text{f}\left(x\right)\text{.}\text{g}}^{\prime }\left(x\right)$
${\left[\text{f}\left(x\right)\text{.}\text{g}\left(x\right)\right]}^{\prime }=\left(6x^2-1\right)\left(x^4+3x\right)+\left(2x^3-x\right)\left(4x^3+3\right)$
${\left[\text{f}\left(x\right)\text{.}\text{g}\left(x\right)\right]}^{\prime }=6x^6+18x^3-x^4-3{x+8x}^6+6x^3-4x^4-3x$
${\left[\text{f}\left(x\right)\text{.}\text{g}\left(x\right)\right]}^{\prime }=14x^6-5x^4+24x^3-6x$