Given \( f(x)=3 x^{2}+3 x-2 \), find the slope of the tangent line of \( f \) at the point whe \( x=-4 \).

To find the slope of the tangent line of the function \( f(x) = 3x^2 + 3x - 2 \) at the point where \( x = -4 \), we first need to determine the derivative of the function, which gives us the slope of the tangent line at any point \( x \). The derivative, \( f'(x) \), is calculated as follows: \[ f'(x) = \frac{d}{dx}(3x^2 + 3x - 2) = 6x + 3. \] Next, we substitute \( x = -4 \) into the derivative to find the slope at that specific point: \[ f'(-4) = 6(-4) + 3 = -24 + 3 = -21. \] Thus, the slope of the tangent line of \( f \) at the point when \( x = -4 \) is \(-21\). Now for some fun facts! Did you know that calculus, the foundation for finding derivatives and slopes, was independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century? It was a revolutionary way to describe change and motion, giving rise to mathematical principles that power much of modern science and engineering today! For a real-world application, think of a car navigating a hilly road. The slope of the tangent line at any given point on the road can tell you how steep the hill is at that moment. If the slope is positive, you're going uphill; if it's negative, you're going downhill. Knowing the slope helps drivers understand how to adjust their speed and gear, making it an essential skill in both math and everyday driving!