Find `dy/dx if x^2-ay+3=0`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x^2 - ay + 3 = 0\), we will follow these steps:

### Step 1: Differentiate the equation with respect to \(x\)
We start with the given equation:
\[ x^2 - ay + 3 = 0 \]
Differentiating both sides with respect to \(x\), we apply the differentiation rules:
- The derivative of \(x^2\) is \(2x\).
- The derivative of \(-ay\) (where \(a\) is a constant) is \(-a \frac{dy}{dx}\).
- The derivative of a constant (3) is 0.

Thus, we have:
\[ \frac{d}{dx}(x^2) - \frac{d}{dx}(ay) + \frac{d}{dx}(3) = 0 \]
This simplifies to:
\[ 2x - a \frac{dy}{dx} + 0 = 0 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate \(\frac{dy}{dx}\):
\[ 2x - a \frac{dy}{dx} = 0 \]
Adding \(a \frac{dy}{dx}\) to both sides gives:
\[ 2x = a \frac{dy}{dx} \]

### Step 3: Solve for \(\frac{dy}{dx}\)
To find \(\frac{dy}{dx}\), we divide both sides by \(a\):
\[ \frac{dy}{dx} = \frac{2x}{a} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{2x}{a} \]

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