2.2.2 Terbitan Pertama Hasil Bahagi Dua Polinomial

2.2.2 Terbitan Pertama Hasil Bahagi Dua Polinomial

Cari hasil bahagi terbitan dengan menggunakan kaedah-kaedah yang berikut:

Kaedah 1: Petua Hasil Bahagi

Contoh 1:

\begin{array}{l} y = \frac{3 x}{2 x - 1}, cari \frac{d y}{d x} \\ \frac{d y}{d x} = \frac{(2 x - 1) (3) - (3 x) (2)}{{(2 x - 1)}^{2}} \\ = \frac{6 x - 3 - 6 x}{{(2 x - 1)}^{2}} \\ = \frac{- 3}{{(2 x - 1)}^{2}} \end{array}

Kaedah 2: (Pembezaan terus)

\begin{array}{l} y = \frac{u (p e n g a n g k a)}{v (p e n y e b u t)} \\ \frac{d y}{d x} = \frac{(\begin{array}{l} s a l i n \\ p e n y e b u t \end{array}) (\begin{array}{l} b e z a k a n \\ p e n g a n g k a \end{array}) - (\begin{array}{l} s a l i n \\ p e n g a n g k a \end{array}) (\begin{array}{l} b e z a k a n \\ p e n y e b u t \end{array})}{{(p e n y e b u t)}^{2}} \end{array}

Contoh 2:

Diberi y = \frac{x^{2}}{2 x + 1}, cari \frac{d y}{d x}

Penyelesaian:

\begin{array}{l} y = \frac{x^{2}}{2 x + 1} \\ \frac{d y}{d x} = \frac{(2 x + 1) (2 x) - x^{2} (2)}{{(2 x + 1)}^{2}} \\ = \frac{4 x^{2} + 2 x - 2 x^{2}}{{(2 x + 1)}^{2}} \\ = \frac{2 x^{2} + 2 x}{{(2 x + 1)}^{2}} \end{array}

Contoh 3:

Diberi bahawa y = \frac{4 x^{3}}{{(5 x + 1)}^{3}}, cari \frac{d y}{d x}

Penyelesaian:

\begin{array}{l} y = \frac{4 x^{3}}{{(5 x + 1)}^{3}} \\ \frac{d y}{d x} = \frac{{(5 x + 1)}^{3} (12 x^{2}) - 4 x^{3} .3 {(5 x + 1)}^{2} .5}{{[{(5 x + 1)}^{3}]}^{2}} \\ = \frac{{(5 x + 1)}^{3} (12 x^{2}) - 60 x^{3} {(5 x + 1)}^{2}}{{(5 x + 1)}^{6}} \\ = \frac{(12 x^{2}) {(5 x + 1)}^{2} [(5 x + 1) - 5 x]}{{(5 x + 1)}^{6}} \\ = \frac{(12 x^{2}) {(5 x + 1)}^{2} (1)}{{(5 x + 1)}^{6}} \\ = \frac{12 x^{2}}{{(5 x + 1)}^{4}} \end{array}