Find \( \frac{d y}{d x} \) by implicit differentiation. \[ x^{2}-10 x y+y^{2}=10 \] \( \frac{d y}{d x}=\square \) Need Help? Read It Watch It Master It

To find \( \frac{dy}{dx} \) using implicit differentiation for the equation \( x^{2} - 10xy + y^{2} = 10 \), we start by differentiating each term with respect to \( x \): 1. The derivative of \( x^{2} \) is \( 2x \). 2. For the term \( -10xy \), we apply the product rule: \( -10(y + x\frac{dy}{dx}) \). 3. The derivative of \( y^{2} \) is \( 2y\frac{dy}{dx} \). 4. The derivative of the constant \( 10 \) is \( 0 \). Putting it all together, we have: \[ 2x - 10\left(y + x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0. \] Now, simplify and solve for \( \frac{dy}{dx} \): \[ 2x - 10y - 10x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0. \] Rearranging gives: \[ (2y - 10x)\frac{dy}{dx} = 10y - 2x. \] Finally, we can solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{10y - 2x}{2y - 10x}. \] Thus, the answer is: \[ \frac{dy}{dx} = \frac{10y - 2x}{2y - 10x}. \]