Mathematics

The slope of a tangent line is given by the derivative, so you're really asking where $\frac{dy}{dx}$ reaches its maximum value. This means you need to find the derivative of the derivative—the second derivative—and set it equal to zero. The key insight is that the steepest tangent occurs where the curve's concavity changes most dramatically, which happens at an inflection point of the original curve.

The slope of a tangent line is given by the derivative, so you're really asking where $\frac{dy}{dx}$ reaches its maximum value. This means you need to find the derivative of the derivative—the second derivative—and set it equal to zero. The key insight is that the steepest tangent occurs where the curve's concavity changes most dramatically, which happens at an inflection point of the original curve.