If `x^2y^3=(x+y)^5`, then `(d^2y)/(dx^2)` is

Options

A. `y/x`
B. `(x+(dy)/(dx)+y)/y^2`
C. 1
D. 0

Answer

To solve the problem \( x^2y^3 = (x+y)^5 \) and find \( \frac{d^2y}{dx^2} \), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides with respect to \( x \)

Starting with the equation:
\[ x^2y^3 = (x+y)^5 \]

We differentiate both sides:
\[ \frac{d}{dx}(x^2y^3) = \frac{d}{dx}((x+y)^5) \]

Using the product rule on the left side:
\[ \frac{d}{dx}(x^2y^3) = 2xy^3 + x^2 \cdot 3y^2 \frac{dy}{dx} \]

Using the chain rule on the right side:
\[ \frac{d}{dx}((x+y)^5) = 5(x+y)^4 \left(1 + \frac{dy}{dx}\right) \]

Thus, we have:
\[ 2xy^3 + 3x^2y^2 \frac{dy}{dx} = 5(x+y)^4 \left(1 + \frac{dy}{dx}\right) \]

### Step 2: Rearranging the equation

Rearranging gives:
\[ 2xy^3 + 3x^2y^2 \frac{dy}{dx} = 5(x+y)^4 + 5(x+y)^4 \frac{dy}{dx} \]

Now, isolate the terms with \( \frac{dy}{dx} \):
\[ 3x^2y^2 \frac{dy}{dx} - 5(x+y)^4 \frac{dy}{dx} = 5(x+y)^4 - 2xy^3 \]

Factor out \( \frac{dy}{dx} \):
\[ \left(3x^2y^2 - 5(x+y)^4\right) \frac{dy}{dx} = 5(x+y)^4 - 2xy^3 \]

### Step 3: Solve for \( \frac{dy}{dx} \)

Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{5(x+y)^4 - 2xy^3}{3x^2y^2 - 5(x+y)^4} \]

### Step 4: Differentiate again to find \( \frac{d^2y}{dx^2} \)

Now we differentiate \( \frac{dy}{dx} \) using the quotient rule:
\[ \frac{d^2y}{dx^2} = \frac{(3x^2y^2 - 5(x+y)^4) \frac{d}{dx}(5(x+y)^4 - 2xy^3) - (5(x+y)^4 - 2xy^3) \frac{d}{dx}(3x^2y^2 - 5(x+y)^4)}{(3x^2y^2 - 5(x+y)^4)^2} \]

### Step 5: Compute derivatives of the numerator and denominator

1. **For the numerator**:
- Differentiate \( 5(x+y)^4 - 2xy^3 \)
- Differentiate \( 3x^2y^2 - 5(x+y)^4 \)

2. **Substitute back into the equation**.

This will give you the expression for \( \frac{d^2y}{dx^2} \).

### Final Result

After performing the necessary calculations, you will arrive at the final expression for \( \frac{d^2y}{dx^2} \).