Insan Pembelajar: October 2019

Turunan Fungsi Aljabar

Rumus-rumus Turunan Fungsi:

$1.\: \: f\left ( x \right )=k; k=konstanta\: sebarang \rightarrow f'\left ( x \right )=0$
   Contoh:
     Tentukan turunan fungsi berikut:
   $1)\: f\left ( x \right )=2 \rightarrow f'\left ( x \right )=0$
     $2)\: f\left ( x \right )=-1 \rightarrow f'\left ( x \right )=0$
     $3)\: f\left ( x \right )=\frac{1}{2} \rightarrow f'\left ( x \right )=0$

$2.\: \: f\left ( x \right )=kx \rightarrow f'\left ( x \right )=k$
     Contoh:
     Tentukan turunan fungsi berikut:
$1)\: f\left ( x \right )=-x \rightarrow f'\left ( x \right )=-1$
      $2)\: f\left ( x \right )=6x \rightarrow f'\left ( x \right )=6$
      $3)\: f\left ( x \right )=\frac{2}{3}x \rightarrow f'\left ( x \right )=\frac{2}{3}$

$3.\: \: f\left ( x \right )=kx^{n} \rightarrow f'\left ( x \right )=n.kx^{n-1}$
     Contoh:
     Tentukan turunan fungsi berikut:
     $1) \: f\left ( x \right )=3x^{2} \rightarrow f'\left ( x \right )=2.3x^{2-1}=6x$
     $2) \: f\left ( x \right )=-2x^{3} \rightarrow f'\left ( x \right )=3.\left (-2 \right )x^{3-1}=-6x^{2}$
     $3)\: f\left ( x \right )=3x^{2}-6x+9\rightarrow f'\left ( x \right )=2.3x^{2}-6+0=6x-6$

$4.\: \: f\left ( x \right )=u.v;\: dimana\: u\: dan\: v\: fungsi\: dalam\: x$
     Contoh:
     Tentukan turunan fungsi berikut:
   $1)\: f\left ( x \right )=\left ( 3x-5 \right )\left ( x^{2}+3x \right )$
          $Misal:\: u=\left ( 3x-5 \right )\rightarrow u'=3$
                           $v=\left ( x^{2}+3x \right )\rightarrow v'=2.x^{2-1}+3=2x+3$
          $f'\left ( x \right )=u'.v+u.v'$
          $f'\left ( x \right )=3.\left ( x^{2}+3x \right )+\left ( 3x-5 \right ).\left ( 2x+3 \right )$
         $f'\left ( x \right )=3x^{2}+9x+6x^{2}+9x-10x-15$
         $f'\left ( x \right )=9x^{2}+8x-15$

     $2)\: f\left ( x \right )=\left ( 2x^{2}-4x \right )\left ( 8-8x^{2} \right )$
          $Misal:\: \: u=\left ( 2x^{2}-4x \right )\rightarrow u'=2.2x^{2-1}-4=4x-4$
                            $v=\left (8- 8x^{2} \right )\rightarrow v'=0-2.8x^{2-1}=-16x$
          $f'\left ( x \right )=u'.v+u.v'$
          $f'\left ( x \right )=\left ( 4x-4 \right ).\left ( 8-8x^{2} \right )+\left ( 2x^{2}-4x \right ).\left ( -16x \right )$
          $f'\left ( x \right )=32x-32x^{3}-32+32x^{2}-32x^{3}+64x^{2}$
          $f'\left ( x \right )=-64x^{3}+96x^{2}+32x-32$

$5.\:\: f\left ( x \right )=\frac{u}{v}\rightarrow f'\left ( x \right )=\frac{u'.v-u.v'}{v^{2}}$
     Contoh:
     Tentukan turunan fungsi berikut:
$1)\: f\left ( x \right )=\frac{3x+5}{4-6x}$
           $Misal\:: u=3x+5\rightarrow u'=3$
                            $v=4-6x\rightarrow v'=-6$
           $f'\left ( x \right )=\frac{u'.v-u.v'}{v^{2}}$
           $f'\left ( x \right )=\frac{3.\left (4-6x \right )-\left (3x+5 \right ).\left ( -6 \right )}{\left ( 4-6x \right )^{2}}$
           $f'\left ( x \right )=\frac{12-18x+6\left ( 3x+5 \right )}{\left ( 4-6x \right )^{2}}$
           $f'\left ( x \right )=\frac{12-18x+18x+30}{\left ( 4-6x \right )^{2}}$
           $f'\left ( x \right )=\frac{42}{\left ( 4-6x \right )^{2}}$

      $2)\: f\left ( x \right )=\frac{x^{2}-1}{6x-5}$
           $Misal:\: u=x^{2}-1\rightarrow u'=2x$
                            $v=6x-5\rightarrow v'=6$
           $f'\left ( x \right )=\frac{u'.v-u.v'}{v^{2}}$
           $f'\left ( x \right )=\frac{2x.\left ( 6x-5 \right )-\left (x^{2}-1 \right ).6}{\left (6x-5 \right )^{2}}$
           $f'\left ( x \right )=\frac{12x^{2}-10x-6x^{2}+6}{\left (6x-5 \right )^{2}}$
           $f'\left ( x \right )=\frac{x^{2}-10x+6}{\left (6x-5 \right )^{2}}$

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