Question

if y = cos²3x, then dy/dx=
a -6 sin 3x cos 3x
b -2 cos 3x
c 2 cos 3x
d 6 cos 3x
e 2 sin 3x cos 3x

Explanation:

Step1: Apply chain - rule

Let $u = \cos 3x$, so $y = u^{2}$. First find $\frac{dy}{du}$ and $\frac{du}{dx}$. The power - rule states that if $y = u^{n}$, then $\frac{dy}{du}=nu^{n - 1}$. For $y = u^{2}$, $\frac{dy}{du}=2u$.

Step2: Differentiate $u=\cos 3x$

Using the chain - rule again, let $v = 3x$, so $u=\cos v$. The derivative of $\cos v$ with respect to $v$ is $-\sin v$, and the derivative of $v = 3x$ with respect to $x$ is 3. Then $\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}=-\sin v\cdot3=- 3\sin 3x$.

Step3: Find $\frac{dy}{dx}$

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=2u$ and $\frac{du}{dx}=-3\sin 3x$ into it. Since $u = \cos 3x$, we have $\frac{dy}{dx}=2\cos 3x\cdot(-3\sin 3x)=-6\sin 3x\cos 3x$.

Answer:

A. $-6\sin 3x\cos 3x$