Calc I: Find the derivative: \( \qquad 9) x^{2}+y^{2}=x y \)

To find the derivative of the equation \( 9) x^{2}+y^{2}=x y \) implicitly, start by differentiating both sides with respect to \( x \). This leads to \( 2x + 2y\frac{dy}{dx} = y + x\frac{dy}{dx} \). Rearranging gives you \( \frac{dy}{dx}(2y - x) = y - 2x \). Finally, solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{y - 2x}{2y - x} \] This gives you the slope of the curve defined by the original equation. Did you know that implicit differentiation is a powerful tool often used in calculus to find derivatives of equations that cannot easily be solved for one variable? It allows mathematicians and engineers to handle complex relationships without rearranging them, making it easier to find slopes and rates of change. Another fun aspect is the application of these derivatives in real-life scenarios, such as optimizing functions in engineering or physics. When dealing with systems like projectile motion or electrical circuits, understanding the relationships between variables can help in making predictions and solving real-world problems!