q11 chain rule: composition of composition\nif ( y=cos ^{2} 3 x ), then ( d y / d x= )\na ( -6 sin 3 x cos 3…

Explanation:

Step1: Let ( u = \cos 3x ), then ( y = u^{2} )

Use the chain rule ( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} ) First, find ( \frac{dy}{du} ): Since ( y = u^{2} ), then ( \frac{dy}{du}=2u )

Step2: Find ( \frac{du}{dx} )

Since ( u=\cos 3x ), let ( v = 3x ), then ( u=\cos v ) By the chain rule ( \frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx} ) ( \frac{du}{dv}=-\sin v ), ( \frac{dv}{dx}=3 ) So ( \frac{du}{dx}=- 3\sin 3x )

Step3: Calculate ( \frac{dy}{dx} )

( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} ) Substitute ( u = \cos 3x ), ( \frac{dy}{du}=2u ) and ( \frac{du}{dx}=-3\sin 3x ) ( \frac{dy}{dx}=2\cos 3x\cdot(-3\sin 3x)=-6\sin 3x\cos 3x )

Answer:

A. -6 sin 3x cos 3x