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$\frac{x}{2}$)] = [1/tan$\frac{x}{2}$] * sec^2$$\frac{x}{2} * $\frac{1}{2}$ = [cos\frac{x/2}{sin}$$\frac{x}{2}] * [1/cos^2$$\frac{x}{2}] * $\frac{1}{2}$ = $\frac{1}{2*sin(x/2}$*cos$\frac{x}{2}$) = 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.