\frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0 \Rightarrow t = -\frac{1}{3}

["# Solving \frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0: Find the Critical Point", "Understanding how to differentiate quadratic functions is essential in calculus, especially when analyzing motion, optimization problems, and rate of change. This article explains step-by-step how to solve the equation (\frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0) and finds the value of (t) where the tangent line to the function is horizontal.", "## Differentiating the Quadratic Expression", "The expression (6t^2 + 4t + 30) represents a quadratic function of (t). To analyze its rate of change at any point, we compute its derivative with respect to (t).", "The derivative of:\n- (6t^2) is (12t) (using the power rule: (\frac{d}{dt}(t^n) = nt^{n-1})).\n- (4t) is (4).\n- The constant (30) vanishes.", "So,\n[\n\frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4\n]", "## Solving (\frac{d}{dt}(6t^2 + 4t + 30) = 0)", "Setting the derivative equal to zero identifies critical points where the slope of the original function is zero — these are potential local maxima, minima, or inflection points.", "[\n12t + 4 = 0\n]", "Solving for (t):", "[\n12t = -4 \quad \Rightarrow \quad t = -\frac{4}{12} = -\frac{1}{3}\n]", "## Interpretation and Significance", "At (t = -\frac{1}{3}), the rate of change of the function (6t^2 + 4t + 30) is zero. This means the tangent line to the curve is horizontal at this point. While the function is a parabola opening upward (since the coefficient of (t^2) is positive), this point represents the minimum value of the quadratic expression.", "In real-world applications, such as motion or cost functions, this critical point could signal the slowest or fastest increase, depending on context — but a flat slope means a temporary stagnation in growth.", "## Summary", "The equation (\frac{d}{dt}(6t^2 + 4t + 30) = 12t + 4 = 0) simplifies algebraically to find (t = -\frac{1}{3}). This tool repeats often in calculus for identifying key features of functions, such as stationary points. Knowing how to compute derivatives and solve these equations is fundamental to understanding dynamic systems.", "---", "Key Takeaway:\n[\n\boxed{\frac{d}{dt}(6t^2 + 4t + 30) = 0 \Rightarrow t = -\frac{1}{3}}\n]\nThis result highlights a fundamental concept linking differentiation to function behavior through critical point analysis."]