Find dy/dx by implicit differentiation for x^2 + y^2 = 25

Problem: Find $\frac{dy}{dx}$ by implicit differentiation for $x^2 + y^2 = 25$.

Steps

1. Differentiate both sides with respect to $x$
   $$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)$$
2. Apply power rule and chain rule
   $$2x + 2y \frac{dy}{dx} = 0$$
3. Subtract $2x$ from both sides
   $$2y \frac{dy}{dx} = -2x$$
4. Divide both sides by $2y$
   $$\frac{dy}{dx} = \frac{-2x}{2y}$$
5. Simplify the fraction
   $$\frac{dy}{dx} = -\frac{x}{y}$$