`5x^2+5y^2-7y+3x=2=0`

Answer

To differentiate the equation \(5x^2 + 5y^2 - 7y + 3x + 2 = 0\) with respect to \(x\), we will follow these steps:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 5x^2 + 5y^2 - 7y + 3x + 2 = 0 \]
Differentiating each term with respect to \(x\):
- The derivative of \(5x^2\) is \(10x\).
- The derivative of \(5y^2\) is \(10y \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(-7y\) is \(-7 \frac{dy}{dx}\).
- The derivative of \(3x\) is \(3\).
- The derivative of the constant \(2\) is \(0\).

Putting it all together, we have:
\[ 10x + 10y \frac{dy}{dx} - 7 \frac{dy}{dx} + 3 + 0 = 0 \]

### Step 2: Rearranging the equation
Now, we can rearrange the equation:
\[ 10x + 3 + (10y - 7) \frac{dy}{dx} = 0 \]

### Step 3: Isolate \(\frac{dy}{dx}\)
Next, we isolate \(\frac{dy}{dx}\):
\[ (10y - 7) \frac{dy}{dx} = -10x - 3 \]
Now, divide both sides by \(10y - 7\):
\[ \frac{dy}{dx} = \frac{-10x - 3}{10y - 7} \]

### Final Answer:
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{-10x - 3}{10y - 7} \]

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