If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$ , then $\frac { d y } { d x } = 2 x + 3 y$ .

Question

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To find $\frac{dy}{dx}$, we need to differentiate both sides of the equation $x^2 + 3xy + y^2 = 15$ with respect to $x$, applying the product rule to $3xy$. Differentiating $x^2$ gives $2x$, for $3xy$ it gives $3y + 3x\frac{dy}{dx}$ (using the product rule), and for $y^2$ it gives $2y\frac{dy}{dx}$ (since $y$ is a function of $x$). Setting the derivative of the constant $15$ to $0$, we get $2x + 3y + 3x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$, which does not simplify to $\frac{dy}{dx} = 2x + 3y$.

If x2+3xy+y2=15x ^ { 2 } + 3 x y + y ^ { 2 } = 15x2+3xy+y2=15

If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$