Find frac{dy}{dx} for the equation x^{2}+xy+y | vedclass

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Question

(A) The given equation is $x^{2}+xy+y^{2}=100$. Differentiating both sides with respect to $x$,we get: $\frac{d}{dx}(x^{2}+xy+y^{2}) = \frac{d}{dx}(10

Vedclass · Accepted Answer

$-\frac{2x+y}{x+2y}$

Vedclass · Answer

(A) The given equation is $x^{2}+xy+y^{2}=100$. Differentiating both sides with respect to $x$,we get: $\frac{d}{dx}(x^{2}+xy+y^{2}) = \frac{d}{dx}(100)$ Using the sum rule and the chain rule: $\frac{d}{dx}(x^{2}) + \frac{d}{dx}(xy) + \frac{d}{dx}(y^{2}) = 0$ Applying the product rule to $xy$: $2x + (y \cdot 1 + x \cdot \frac{dy}{dx}) + 2y \frac{dy}{dx} = 0$ Grouping the $\frac{dy}{dx}$ terms: $2x + y + (x + 2y) \frac{dy}{dx} = 0$ Solving for $\frac{dy}{dx}$: $(x + 2y) \frac{dy}{dx} = -(2x + y)$ $\frac{dy}{dx} = -\frac{2x+y}{x+2y}$

Find $\frac{dy}{dx}$ for the equation $x^{2}+xy+y^{2}=100$.

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