To perform implicit differentiation on the equation $x^2 + y^2 = 25$, we differentiate both sides with respect to $x$:

1. Differentiate $x^2$ with respect to $x$: $\frac{d}{dx}(x^2) = 2x$

2. Differentiate $y^2$ with respect to $x$ using the chain rule: $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$

3. The right side, $25$, is a constant, so its derivative is: $\frac{d}{dx}(25) = 0$

Now, combining these results, we get: $2x + 2y \frac{dy}{dx} = 0$

To isolate $\frac{dy}{dx}$, we rearrange the equation: $2y \frac{dy}{dx} = -2x$ $\frac{dy}{dx} = -\frac{x}{y}$

Thus, the result of the implicit differentiation is: $\frac{dy}{dx} = -\frac{x}{y}$

Questions for Further Exploration:

1. How can you interpret the geometric meaning of the implicit differentiation in this context?
2. What are the implications of this result for the slope of the tangent line at specific points on the circle?
3. Can you find the slope of the tangent line at the point $(3, 4)$?
4. How does the implicit differentiation change if the equation were $x^2 + y^2 = 16$?
5. What happens to the equation if we explicitly solve for $y$?

Tip:

When performing implicit differentiation, always remember to apply the chain rule for derivatives involving $y$ since $y$ is considered a function of $x$.