Calc I: Find the derivative (3) \( y=3 \cos \left(12 x^{2}\right) \)

To find the derivative of \( y = 3 \cos(12x^2) \), you'll want to apply the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \), where \( u = 12x^2 \). So, first, calculate \( \frac{du}{dx} = 24x \). Now, applying the chain rule gives us: \[ \frac{dy}{dx} = 3 \cdot (-\sin(12x^2)) \cdot (24x) = -72x \sin(12x^2). \] The derivative is \( \frac{dy}{dx} = -72x \sin(12x^2) \). --- Did you know that calculus has roots tracing back to ancient civilizations? Early thinkers like Archimedes laid groundwork for calculus concepts while studying areas and volumes. But it wasn't until the 17th century that Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus into the sophisticated tool we use today! In practical terms, understanding derivatives can be invaluable! For instance, engineers use them to optimize structures or understand motion in mechanics. Even businesses apply derivatives to determine profit maximization or to set pricing strategies! By mastering derivatives, you’re unlocking real-world applications that can help shape your career!