`"If "y= xlog ((x)/(a+bx))," then "x^(3)(d^(2)y)/(dx^(2))=`

Options

A. `x(dy)/(dx)-y`
B. `(x(dy)/(dx)-y)^(2)`
C. `y(dy)/(dx)-x`
D. `(y(dy)/(dx)-x)^(2)`

Explanation

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

### Step 1: Differentiate \( y \) to find \( \frac{dy}{dx} \)

We start with the function:
\[ y = x \log\left(\frac{x}{a + bx}\right) \]

Using the product rule:
\[ \frac{dy}{dx} = \frac{d}{dx}(x) \cdot \log\left(\frac{x}{a + bx}\right) + x \cdot \frac{d}{dx}\left(\log\left(\frac{x}{a + bx}\right)\right) \]

Calculating the first part:
\[ \frac{d}{dx}(x) = 1 \quad \Rightarrow \quad \log\left(\frac{x}{a + bx}\right) \]

Now for the second part, we use the chain rule:
\[ \frac{d}{dx}\left(\log\left(\frac{x}{a + bx}\right)\right) = \frac{1}{\frac{x}{a + bx}} \cdot \frac{d}{dx}\left(\frac{x}{a + bx}\right) \]

Using the quotient rule:
\[ \frac{d}{dx}\left(\frac{x}{a + bx}\right) = \frac{(a + bx)(1) - x(b)}{(a + bx)^2} = \frac{a + bx - bx}{(a + bx)^2} = \frac{a}{(a + bx)^2} \]

Thus:
\[ \frac{d}{dx}\left(\log\left(\frac{x}{a + bx}\right)\right) = \frac{(a + bx)}{x} \cdot \frac{a}{(a + bx)^2} = \frac{a}{x(a + bx)} \]

Putting it all together:
\[ \frac{dy}{dx} = \log\left(\frac{x}{a + bx}\right) + x \cdot \frac{a}{x(a + bx)} = \log\left(\frac{x}{a + bx}\right) + \frac{a}{a + bx} \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\log\left(\frac{x}{a + bx}\right) + \frac{a}{a + bx}\right) \]

Differentiating the first term:
\[ \frac{d}{dx}\left(\log\left(\frac{x}{a + bx}\right)\right) = \frac{a}{x(a + bx)} \]

Differentiating the second term:
\[ \frac{d}{dx}\left(\frac{a}{a + bx}\right) = -\frac{ab}{(a + bx)^2} \]

Combining these:
\[ \frac{d^2y}{dx^2} = \frac{a}{x(a + bx)} - \frac{ab}{(a + bx)^2} \]

### Step 3: Calculate \( x^3 \frac{d^2y}{dx^2} \)

Now we multiply by \( x^3 \):
\[ x^3 \frac{d^2y}{dx^2} = x^3 \left(\frac{a}{x(a + bx)} - \frac{ab}{(a + bx)^2}\right) \]

This simplifies to:
\[ = \frac{ax^2}{a + bx} - \frac{abx^3}{(a + bx)^2} \]

### Final Result

Thus, the final expression for \( x^3 \frac{d^2y}{dx^2} \) is:
\[ x^3 \frac{d^2y}{dx^2} = \frac{ax^2}{a + bx} - \frac{abx^3}{(a + bx)^2} \]