1. The problem is to find $\frac{dy}{dx}$ by implicit differentiation for the equation $$x^2 + y^2 = 100.$$\n\n2. The formula used is implicit differentiation, which means we differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$ (i.e., $y = y(x)$). Remember, when differentiating terms involving $y$, use the chain rule: $\frac{d}{dx}[y^n] = n y^{n-1} \frac{dy}{dx}$.\n\n3. Differentiate both sides: $$\frac{d}{dx}[x^2] + \frac{d}{dx}[y^2] = \frac{d}{dx}[100].$$\n\n4. Calculate each derivative: $$2x + 2y \frac{dy}{dx} = 0.$$\n\n5. Solve for $\frac{dy}{dx}$: $$2y \frac{dy}{dx} = -2x \implies \frac{dy}{dx} = \frac{-2x}{2y} = \frac{-x}{y}.$$\n\n6. Therefore, the derivative $\frac{dy}{dx}$ is $$\boxed{\frac{dy}{dx} = \frac{-x}{y}}.$$\n\nThis means the slope of the tangent line to the circle $x^2 + y^2 = 100$ at any point $(x,y)$ is $-x/y$.