The chain rule

A composite f = v∘u is built by putting t = u(x) and then f = v(t). The chain rule says df/dx = (dv/dt)·(dt/dx), whenever both derivatives exist.

Worked example: (2x + 1)³. Expanding gives 8x³ + 12x² + 6x + 1 with derivative 24x² + 24x + 6. The chain rule gets the same, 3(2x + 1)² × 2 = 6(2x + 1)², without expanding, which matters for a power like (2x + 1)¹⁰⁰.

Read it as outside-in: differentiate the outer function at the inner one, then multiply by the derivative of the inner one.

Worked example: sin(x²). With t = x², d/dx = cos t × 2x = 2x cos(x²).

Longer chains multiply every link: if f = w(u(v(x))), with t = v(x) and s = u(t), then df/dx = (dw/ds)(ds/dt)(dt/dx).

Worked example: tan(2x + 3). The outer derivative is sec²(2x + 3) and the inner one is 2, so the answer is 2 sec²(2x + 3).

Worked example: cos(√x). The chain is cos, then square root: −sin(√x) × 1/(2√x).