17. \( \left(\sin ^{2} x-\cos ^{2} x\right)^{\prime}= \)

To find the derivative of \( \left(\sin^{2} x - \cos^{2} x\right) \), we can apply the differentiation rules. The derivative of \( \sin^2 x \) is \( 2 \sin x \cos x \) (using the chain rule), and the derivative of \( -\cos^2 x \) is \( 2 \cos x \sin x \). Therefore, \[ \left(\sin^{2} x - \cos^{2} x\right)^{\prime} = 2\sin x \cos x - 2\cos x \sin x = 0. \] So, the derivative is \( 0 \). For a more practical understanding, consider the trigonometric identity that relates square terms. Recognizing that the expression \( \sin^2 x - \cos^2 x \) can be transformed to \( -\cos(2x) \) through the identity \( \sin^2 x - \cos^2 x = -\cos(2x) \) can simplify things. Hence, the derivative simply becomes \( \sin(2x) \), doubling the angle! Ever wondered where those identities came from? Delving into the history of trigonometric identities provides interesting insights! The origins trace back to Ancient Greek mathematicians who explored circles, angles, and periodic functions. They developed these relationships long before calculators or computers, creating a fascinating blend of art and science that laid the foundation for modern mathematics. If you're looking to delve deeper into the practical applications of these derivatives, look no further than physics and engineering! The oscillator equations in physics often use these sin and cos functions to describe wave motion. From sound waves to light, understanding these derivatives helps us model real-world phenomena. So, next time you hear a note, think of the math dancing in the background!