The slope of tangent to the curve `y = 2 sin^(2)(3x)` at `x = (pi)/(4)` is :

Options

A. `(1)/(sqrt(2))`
B. `sqrt(2)`
C. `-6`
D. `-1`

Answer

To find the slope of the tangent to the curve \( y = 2 \sin^2(3x) \) at \( x = \frac{\pi}{4} \), we will follow these steps:

### Step 1: Differentiate the function
We start with the function:
\[ y = 2 \sin^2(3x) \]
To find the slope of the tangent, we need to differentiate \( y \) with respect to \( x \). We will use the chain rule and the power rule for differentiation.

Using the chain rule:
\[ \frac{dy}{dx} = 2 \cdot 2 \sin(3x) \cdot \frac{d}{dx}(\sin(3x)) \]
Now, we differentiate \( \sin(3x) \):
\[ \frac{d}{dx}(\sin(3x)) = \cos(3x) \cdot \frac{d}{dx}(3x) = 3 \cos(3x) \]
Thus, substituting back, we have:
\[ \frac{dy}{dx} = 4 \sin(3x) \cdot 3 \cos(3x) = 12 \sin(3x) \cos(3x) \]

### Step 2: Use the double angle identity
We can simplify \( 12 \sin(3x) \cos(3x) \) using the double angle identity:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
This gives us:
\[ \frac{dy}{dx} = 6 \sin(6x) \]

### Step 3: Evaluate the derivative at \( x = \frac{\pi}{4} \)
Now we need to find the slope at \( x = \frac{\pi}{4} \):
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = 6 \sin\left(6 \cdot \frac{\pi}{4}\right) = 6 \sin\left(\frac{3\pi}{2}\right) \]
We know that:
\[ \sin\left(\frac{3\pi}{2}\right) = -1 \]
Thus:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = 6 \cdot (-1) = -6 \]

### Final Answer
The slope of the tangent to the curve at \( x = \frac{\pi}{4} \) is:
\[ \boxed{-6} \]