Cue Column:

• What is the chain rule formula?
• How to handle triple composition?
• Common mistakes?

Notes Column: The chain rule states: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Think of it as: "derivative of the outer function evaluated at the inner function, TIMES derivative of the inner function."

For triple composition y = f(g(h(x))): dy/dx = f'(g(h(x))) * g'(h(x)) * h'(x)

Example: d/dx[sin(e^(x^2))] = cos(e^(x^2)) * e^(x^2) * 2x (outer = sin, middle = e^t, inner = x^2)

Example: d/dx[ln(tan(x/2))] = [1/tan(x/2)] * sec^2(x/2) * (1/2) = [cos(x/2)/sin(x/2)] * [1/cos^2(x/2)] * (1/2) = 1/(2*sin(x/2)*cos(x/2)) = 1/sin(x) = cosec(x)

Common mistake: Forgetting the inner derivative. d/dx[sin(3x)] = cos(3x) * 3, NOT cos(3x).

Summary: Chain rule is just "multiply by the derivative of whatever is inside." Apply it layer by layer from outside to inside.