Bab 9 Pembezaan

Bab 9 Pembezaan

9.2.1 Terbitan Pertama untuk Fungsi Polinomial (Contoh)

Contoh:
Cari dy/dbagi setiap fungsi yang berikut:
(a) y = 12
(b) y = x4
(c) y = 3x
(d) y = 5x3
$\begin{array}{l}\text{(e)}y=\frac{1}{x}\\ \text{(f)}y=\frac{2}{{x}^{4}}\\ \text{(g)}y=\frac{2}{5{x}^{2}}\\ \text{(h)}y=\text{3}\sqrt{x}\\ \text{(i)}y=4\sqrt{{x}^{3}}\end{array}$

Penyelesaian
:
(a) y = 12
dy/d= 0

(b)
y = x4
dy/d= 4x3

(c)
y = 3x
dy/d= 3

(d)
y = 5x3
dy/d= 15x2

(e)
$\begin{array}{l}y=\frac{1}{x}={x}^{-1}\\ \frac{dy}{dx}=-{x}^{-1-1}=-\frac{1}{{x}^{2}}\end{array}$

(f)
$\begin{array}{l}y=\frac{2}{{x}^{4}}=2{x}^{-4}\\ \frac{dy}{dx}=-4\left(2{x}^{-4-1}\right)\\ \text{}=-8{x}^{-5}=-\frac{8}{{x}^{5}}\end{array}$

(g)
$\begin{array}{l}y=\frac{2}{5{x}^{2}}=\frac{2{x}^{-2}}{5}\\ \frac{dy}{dx}=-2\left(\frac{2{x}^{-2-1}}{5}\right)\\ \text{}=-\frac{4{x}^{-3}}{5}\\ \text{}=-\frac{4}{5{x}^{3}}\end{array}$

(h)
$\begin{array}{l}y=\text{3}\sqrt{x}=3{\left(x\right)}^{\frac{1}{2}}\\ \frac{dy}{dx}=\frac{1}{2}\left(3{x}^{\frac{1}{2}-1}\right)\\ \text{}=\frac{3}{2}{x}^{-\frac{1}{2}}=\frac{3}{2\sqrt{x}}\end{array}$

(i)
$\begin{array}{l}y=4\sqrt{{x}^{3}}=4{\left({x}^{3}\right)}^{\frac{1}{2}}=4{x}^{\frac{3}{2}}\\ \frac{dy}{dx}=\frac{3}{2}\left(4{x}^{\frac{3}{2}-1}\right)\\ \text{}=6{x}^{\frac{1}{2}}=6\sqrt{x}\end{array}$