3.4.1—Advanced differentiation

For a composite y=f(g(x)), y′=f′(g(x))g′(x). Exponential, logarithmic and trigonometric outer functions all use this chain-rule structure.

Label the inner function and its derivative explicitly. For products or quotients, combine the product/quotient rule with the chain rule rather than expanding unnecessarily.

d[sin((2x+1)²)]/dx=4(2x+1)cos((2x+1)²).

The inner derivative cannot be omitted just because the outer derivative looks familiar.