`ax^2+2hxy+by^2+2gx+2fy+c=0`

Answer

To differentiate the given equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) with respect to \( x \), we will follow these steps:

### Step 1: Differentiate each term with respect to \( x \)

We will apply the differentiation rules to each term in the equation. The differentiation of each term is as follows:

1. \( \frac{d}{dx}(ax^2) = 2ax \)
2. \( \frac{d}{dx}(2hxy) = 2hy + 2hx\frac{dy}{dx} \) (using the product rule)
3. \( \frac{d}{dx}(by^2) = 2by\frac{dy}{dx} \) (since \( y \) is a function of \( x \))
4. \( \frac{d}{dx}(2gx) = 2g \)
5. \( \frac{d}{dx}(2fy) = 2f\frac{dy}{dx} \) (again, since \( y \) is a function of \( x \))
6. \( \frac{d}{dx}(c) = 0 \)

Putting this all together, we have:

\[ 2ax + (2hy + 2hx\frac{dy}{dx}) + (2by\frac{dy}{dx}) + 2g + (2f\frac{dy}{dx}) + 0 = 0 \]

### Step 2: Combine like terms

Now we can combine the terms that contain \( \frac{dy}{dx} \):

\[ 2ax + 2g + 2hy + (2hx + 2by + 2f)\frac{dy}{dx} = 0 \]

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

To isolate \( \frac{dy}{dx} \), we move the other terms to the right side of the equation:

\[ (2hx + 2by + 2f)\frac{dy}{dx} = - (2ax + 2g + 2hy) \]

### Step 4: Solve for \( \frac{dy}{dx} \)

Now, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = -\frac{2ax + 2g + 2hy}{2hx + 2by + 2f} \]

### Step 5: Simplify the expression

We can simplify the expression by factoring out 2 from the numerator and denominator:

\[ \frac{dy}{dx} = -\frac{ax + g + hy}{hx + by + f} \]

### Final Answer

Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = -\frac{ax + hy + g}{hx + by + f} \]

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