(32) If $$ y=\cos (\pi-3 x)^{2} $$, then $$ \frac{d y}{d x}= $$ $$ \begin{array}{ll} ext { (a) }-6 \cos (\pi-3 x) & ext { (b) }-\sin (\pi-3 x)^{2}+2(\pi-3 x) \cos (\pi-3 x)^{2} \ ext { (c) } 6(\pi-3 x) \sin (\pi-3 x)^{2} & ext { (d) }-2 \sin (\pi-3 x)^{2}\end{array} $$

Question

(32) If $$ y=\cos (\pi-3 x)^{2} $$, then $$ \frac{d y}{d x}= $$ $$ \begin{array}{ll}	ext { (a) }-6 \cos (\pi-3 x) & 	ext { (b) }-\sin (\pi-3 x)^{2}+2(\pi-3 x) \cos (\pi-3 x)^{2} \ 	ext { (c) } 6(\pi-3 x) \sin (\pi-3 x)^{2} & 	ext { (d) }-2 \sin (\pi-3 x)^{2}\end{array} $$

UpStudy AI · Accepted Answer

To find $$\frac{dy}{dx}$$ for the function $$y = \cos((\pi - 3x)^2)$$, we will use the chain rule. The chain rule states that if you have a composite function $$y = f(g(x))$$, then the derivative is given by:

$$
\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
$$

### Step 1: Identify the outer and inner functions
In our case:
- The outer function $$f(u) = \cos(u)$$ where $$u = (\pi - 3x)^2$$.
- The inner function $$g(x) = (\pi - 3x)^2$$.

### Step 2: Differentiate the outer function
The derivative of the outer function $$f(u) = \cos(u)$$ is:
$$
f'(u) = -\sin(u)
$$

### Step 3: Differentiate the inner function
Now we differentiate the inner function $$g(x) = (\pi - 3x)^2$$ using the chain rule again:
$$
g'(x) = 2(\pi - 3x) \cdot (-3) = -6(\pi - 3x)
$$

### Step 4: Apply the chain rule
Now we can apply the chain rule:
$$
\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = -\sin((\pi - 3x)^2) \cdot (-6(\pi - 3x))
$$

This simplifies to:
$$
\frac{dy}{dx} = 6(\pi - 3x) \sin((\pi - 3x)^2)
$$

### Final Answer
Thus, the derivative $$\frac{dy}{dx}$$ is:
$$
\frac{dy}{dx} = 6(\pi - 3x) \sin((\pi - 3x)^2)
$$

Looking at the options provided, the correct answer is:
**(c) $$6(\pi - 3x) \sin((\pi - 3x)^2)$$**.
The derivative $$ \frac{dy}{dx} $$ is $$ 6(\pi - 3x) \sin((\pi - 3x)^2) $$.

(32) If \( y=\cos (\pi-3 x)^{2} \), then \( \frac{d y}{d x}= \) \( \begin{array}{ll}\text { (a) }-6 \cos (\pi-3 x) & \text { (b) }-\sin (\pi-3 x)^{2}+2(\pi-3 x) \cos (\pi-3 x)^{2} \\ \text { (c) } 6(\pi-3 x) \sin (\pi-3 x)^{2} & \text { (d) }-2 \sin (\pi-3 x)^{2}\end{array} \)

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