2.4.3—Differentiation rules

A composite expression may require chain, product, quotient or implicit differentiation. Read its structure before applying a familiar rule.

Rewrite constants and powers clearly, differentiate one layer at a time, and check the result by estimating the sign or scale of the gradient at a simple point.

For y=(x²+1)^4/(x−1), treat the numerator and denominator as factors and apply product/quotient plus chain rules rather than expanding blindly.

There is no single “power rule” shortcut for a quotient or a composite power without the required extra factors.