If `y = x^(2). e^(xy)` then derivative of y is

Options

A. `- (y)/(x) ((2 + xy))/((1 - xy))`
B. `(y)/(x) ((2+ xy)/(1 - xy))`
C. `(x(2 + xy))/(y(1-xy))`
D. None of these

Explanation

To find the derivative of the function \( y = x^2 e^{xy} \), we will use the product rule and implicit differentiation. Here’s the step-by-step solution:

### Step 1: Identify the components
The function \( y \) can be expressed as a product of two functions:
- \( u = x^2 \)
- \( v = e^{xy} \)

### Step 2: Apply the product rule
The product rule states that if \( y = uv \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step 3: Differentiate \( u \) and \( v \)
1. Differentiate \( u = x^2 \):
\[ \frac{du}{dx} = 2x \]

2. Differentiate \( v = e^{xy} \) using the chain rule:
- Let \( z = xy \), then \( v = e^z \).
- The derivative of \( v \) is:
\[ \frac{dv}{dx} = e^{xy} \cdot \frac{d(xy)}{dx} \]
- Now, apply the product rule to \( xy \):
\[ \frac{d(xy)}{dx} = x \frac{dy}{dx} + y \]
- Therefore,
\[ \frac{dv}{dx} = e^{xy} (x \frac{dy}{dx} + y) \]

### Step 4: Substitute into the product rule formula
Now substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) back into the product rule:
\[ \frac{dy}{dx} = x^2 \cdot e^{xy} (x \frac{dy}{dx} + y) + e^{xy} \cdot 2x \]

### Step 5: Simplify the equation
Factor out \( e^{xy} \):
\[ \frac{dy}{dx} = e^{xy} \left( x^2 (x \frac{dy}{dx} + y) + 2x \right) \]
This simplifies to:
\[ \frac{dy}{dx} = e^{xy} \left( x^3 \frac{dy}{dx} + x^2 y + 2x \right) \]

### Step 6: Isolate \( \frac{dy}{dx} \)
Rearranging gives:
\[ \frac{dy}{dx} - e^{xy} x^3 \frac{dy}{dx} = e^{xy} (x^2 y + 2x) \]
Factor out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} (1 - e^{xy} x^3) = e^{xy} (x^2 y + 2x) \]
Thus,
\[ \frac{dy}{dx} = \frac{e^{xy} (x^2 y + 2x)}{1 - e^{xy} x^3} \]

### Step 7: Substitute \( e^{xy} \)
Since \( e^{xy} = \frac{y}{x^2} \), substitute this into the equation:
\[ \frac{dy}{dx} = \frac{\frac{y}{x^2} (x^2 y + 2x)}{1 - \frac{y}{x^2} x^3} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{y (x^2 y + 2x)}{x^2 - y x} \]

### Final Answer
Thus, the derivative of \( y \) is:
\[ \frac{dy}{dx} = \frac{y (x^2 y + 2x)}{x^2 - y x} \]