The slope of tangent to the curve `y= cos^2` (3x) at `x= pi/6` is

Options

A. 0
B. 6
C. `-6`
D. `pi`

Answer

To find the slope of the tangent to the curve \( y = \cos^2(3x) \) at \( x = \frac{\pi}{6} \), we need to follow these steps:

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

Using the chain rule:
\[ \frac{dy}{dx} = 2\cos(3x) \cdot \frac{d}{dx}(\cos(3x)) \]

### Step 2: Differentiate \( \cos(3x) \)
Now we differentiate \( \cos(3x) \):
\[ \frac{d}{dx}(\cos(3x)) = -\sin(3x) \cdot \frac{d}{dx}(3x) = -3\sin(3x) \]

### Step 3: Substitute back into the derivative
Now substituting this back into our derivative:
\[ \frac{dy}{dx} = 2\cos(3x) \cdot (-3\sin(3x)) = -6\cos(3x)\sin(3x) \]

### Step 4: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = -6\cos(3x)\sin(3x) \]

### Step 5: Evaluate the derivative at \( x = \frac{\pi}{6} \)
Now we need to evaluate this derivative at \( x = \frac{\pi}{6} \):
\[ \frac{dy}{dx} \bigg|_{x=\frac{\pi}{6}} = -6\cos\left(3 \cdot \frac{\pi}{6}\right)\sin\left(3 \cdot \frac{\pi}{6}\right) \]
Calculating \( 3 \cdot \frac{\pi}{6} = \frac{\pi}{2} \):
\[ = -6\cos\left(\frac{\pi}{2}\right)\sin\left(\frac{\pi}{2}\right) \]

### Step 6: Substitute the values of sine and cosine
Now we substitute the values:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus:
\[ \frac{dy}{dx} \bigg|_{x=\frac{\pi}{6}} = -6 \cdot 0 \cdot 1 = 0 \]

### Conclusion
The slope of the tangent to the curve \( y = \cos^2(3x) \) at \( x = \frac{\pi}{6} \) is:
\[ \boxed{0} \]