What is the chain rule and how do I differentiate sin(3x² + 1)?

The chain rule handles composite functions, where one function sits inside another. It says: differentiate the outer function keeping the inside unchanged, then multiply by the derivative of the inside. For y = sin(3x² + 1), the outer function is sine and the inside is u = 3x² + 1. The derivative of sin u with respect to u is cos u, and the derivative of the inside 3x² + 1 is 6x. Multiply them: the answer is cos(3x² + 1) times 6x, that is 6x cos(3x² + 1). A useful way to remember is dy/dx = dy/du times du/dx. Always work from the outside inward, and keep peeling layers until you reach x. With practice this becomes automatic.