The chain rule, product rule, and quotient rule all reduce differentiation to multiplication—they just tell you which pieces to multiply and in what order. Drag the slider to see each rule build its derivative step by step.
The chain rule differentiates composite functions where one function is nested inside another. For y = f(g(x)), the derivative dy/dx = f'(g(x)) × g'(x) multiplies the outer function's derivative (evaluated at the inner function) by the inner function's derivative. This appears in functions like (3x² + 1)⁵, where the outer function is ( )⁵ and the inner is 3x² + 1. The chain rule tells us to take 5(3x² + 1)⁴ and multiply by 6x, giving 30x(3x² + 1)⁴. Without the chain rule, differentiating nested exponentials, trigonometric compositions, and logarithms would be impossible. Every time you see a function inside another function, the chain rule is the only tool that works.