Implicit differentiation to find dy/dx at a point

Differentiate both sides of $x^2 + y^2 = 25$ with respect to $x$. The left side gives $2x + 2y\frac{dy}{dx}$ (using the chain rule on $y^2$), and the right side, being constant, gives $0$. This yields $2x + 2y\frac{dy}{dx} = 0$.

Solve for $\frac{dy}{dx}$: subtract $2x$ from both sides and divide by $2y$ to get $\frac{dy}{dx} = -\frac{x}{y}$. Substituting $x=3$ and $y=4$ gives $\frac{dy}{dx} = -\frac{3}{4}$.

The derivative formula found earlier is negative x over y. At the point negative three, four, substituting gives negative negative three over four, which simplifies to positive three fourths. So the slope at this point is positive.