\( 10 \quad \) b) Diberi \( y=\frac{x}{3 x-5} \) Tunjukkan \( \frac{d y}{d x}=-\frac{5}{(3 x-5)^{2}} \) Seterusnya, can \( \int \frac{1}{(3 x-5)^{2}} d x \) us \( \quad \frac{x}{3 x-5} \). Show that \( \frac{d y}{d x}=-\frac{5}{(3 x-5)^{2}} \) Given Hence. find \( \int \frac{1}{(3 x-5)^{2}} d x \)

To find \( \frac{dy}{dx} \) for the function \( y = \frac{x}{3x - 5} \), we can use the quotient rule, which states that if \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{u'v - uv'}{v^2} \). For this case, let \( u = x \) and \( v = 3x - 5 \). Thus, \( u' = 1 \) and \( v' = 3 \). Applying the quotient rule: \[ \frac{dy}{dx} = \frac{(1)(3x - 5) - (x)(3)}{(3x - 5)^2} = \frac{3x - 5 - 3x}{(3x - 5)^2} = \frac{-5}{(3x - 5)^2} \] The derivative is indeed \( \frac{dy}{dx} = -\frac{5}{(3x - 5)^2} \). Now, to integrate \( \int \frac{1}{(3x - 5)^2} \, dx \), we can use a substitution. Let \( u = 3x - 5 \), then \( du = 3dx \) or \( dx = \frac{du}{3} \). The integral transforms into: \[ \int \frac{1}{u^2} \cdot \frac{du}{3} = \frac{1}{3} \int u^{-2} \, du = \frac{1}{3} \left(-\frac{1}{u}\right) + C = -\frac{1}{3(3x - 5)} + C \] Thus, the final answer for the integral is \( -\frac{1}{3(3x - 5)} + C \).