Derivatives of Algebraic Functions Using Chain Rule

The chain rule is the tool you reach for when a function is built inside another — when you have a function of a function, like $(3x+1)^5$ or $\sqrt{x^2+4}$. Instead of expanding everything, the chain rule lets you differentiate layer by layer: first the outer function, then multiply by the derivative of the inner function. This idea is the backbone of almost every derivative you'll compute from here on, and it turn …