Find `dy/dx if x^2+xy+y^2=100`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x^2 + xy + y^2 = 100\), we will use implicit differentiation. Here’s the step-by-step solution:

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

We start with the equation:
\[ x^2 + xy + y^2 = 100 \]

Differentiating each term with respect to \(x\):
- The derivative of \(x^2\) is \(2x\).
- For the term \(xy\), we use the product rule:
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \frac{d}{dx}(x) = x \frac{dy}{dx} + y \]
- The derivative of \(y^2\) is:
\[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
- The derivative of the constant \(100\) is \(0\).

Putting it all together, we have:
\[ 2x + \left(x \frac{dy}{dx} + y\right) + 2y \frac{dy}{dx} = 0 \]

### Step 2: Combine like terms

Now, we can combine the terms involving \(\frac{dy}{dx}\):
\[ 2x + y + \left(x + 2y\right) \frac{dy}{dx} = 0 \]

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

Next, we isolate \(\frac{dy}{dx}\):
\[ \left(x + 2y\right) \frac{dy}{dx} = - (2x + y) \]

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

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

### Final Answer

Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{- (2x + y)}{x + 2y} \]

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