If ` y = x log ((x)/(a + bx)), " then " (x^(3) d^(2) y)/(ax^(2))` is equal to

Options

A. `x(dy)/(dx) - y`
B. `(x (dy)/(dx) - y)^(2)`
C. `y (dy)/(dx) - x `
D. None of these

Explanation

To solve the problem, we need to find the expression for \( \frac{x^3 \frac{d^2 y}{dx^2}}{a x^2} \) given that \( y = x \log\left(\frac{x}{a + bx}\right) \).

### Step 1: Rewrite the function
Start with the given function:
\[ y = x \log\left(\frac{x}{a + bx}\right) \]
Using the properties of logarithms, we can rewrite this as:
\[ y = x \left( \log(x) - \log(a + bx) \right) \]

### Step 2: Differentiate \( y \) with respect to \( x \)
Using the product rule:
\[ \frac{dy}{dx} = \frac{d}{dx}\left(x \log(x)\right) - \frac{d}{dx}\left(x \log(a + bx)\right) \]
For the first term, apply the product rule:
\[ \frac{d}{dx}(x \log(x)) = \log(x) + 1 \]
For the second term, again apply the product rule:
\[ \frac{d}{dx}(x \log(a + bx)) = \log(a + bx) + \frac{bx}{a + bx} \]
Thus,
\[ \frac{dy}{dx} = \log(x) + 1 - \left( \log(a + bx) + \frac{bx}{a + bx} \right) \]

### Step 3: Simplify \( \frac{dy}{dx} \)
Combining the terms we have:
\[ \frac{dy}{dx} = \log(x) - \log(a + bx) + 1 - \frac{bx}{a + bx} \]

### Step 4: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)
Now, we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\log(x) - \log(a + bx) + 1 - \frac{bx}{a + bx}\right) \]
This gives:
\[ \frac{d^2y}{dx^2} = \frac{1}{x} - \frac{b}{a + bx} - \frac{b(a + bx) - bx \cdot b}{(a + bx)^2} \]
Simplifying this expression will yield \( \frac{d^2y}{dx^2} \).

### Step 5: Substitute \( \frac{d^2y}{dx^2} \) into \( \frac{x^3 \frac{d^2y}{dx^2}}{a x^2} \)
Now, we need to multiply \( \frac{d^2y}{dx^2} \) by \( x^3 \) and divide by \( ax^2 \):
\[ \frac{x^3 \frac{d^2y}{dx^2}}{a x^2} = \frac{x \frac{d^2y}{dx^2}}{a} \]

### Step 6: Final expression
After substituting and simplifying, we can find the final expression for the given problem.