If `y=cosx.cos4x.cos8x`, then what is `(1)/(y)(dy)/(dx)` at `x=(pi)/(4)` equal to?

Options

A. `-1`
B. 0
C. 1
D. 3

Answer

To solve the problem where \( y = \cos x \cdot \cos 4x \cdot \cos 8x \) and we need to find \( \frac{1}{y} \cdot \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \), we can follow these steps:

### Step 1: Differentiate \( y \)
We start with the expression for \( y \):
\[ y = \cos x \cdot \cos 4x \cdot \cos 8x \]
To differentiate \( y \), we can use the product rule. The derivative of \( y \) is:
\[ \frac{dy}{dx} = \frac{d}{dx}(\cos x) \cdot \cos 4x \cdot \cos 8x + \cos x \cdot \frac{d}{dx}(\cos 4x) \cdot \cos 8x + \cos x \cdot \cos 4x \cdot \frac{d}{dx}(\cos 8x) \]

### Step 2: Calculate the derivatives
Using the derivative of cosine:
\[ \frac{d}{dx}(\cos kx) = -k \sin kx \]
we can find each term:
1. \( \frac{d}{dx}(\cos x) = -\sin x \)
2. \( \frac{d}{dx}(\cos 4x) = -4 \sin 4x \)
3. \( \frac{d}{dx}(\cos 8x) = -8 \sin 8x \)

Substituting these into the derivative:
\[ \frac{dy}{dx} = (-\sin x) \cdot \cos 4x \cdot \cos 8x + \cos x \cdot (-4 \sin 4x) \cdot \cos 8x + \cos x \cdot \cos 4x \cdot (-8 \sin 8x) \]

### Step 3: Simplify \( \frac{dy}{dx} \)
This can be rewritten as:
\[ \frac{dy}{dx} = -\sin x \cos 4x \cos 8x - 4 \cos x \sin 4x \cos 8x - 8 \cos x \cos 4x \sin 8x \]

### Step 4: Evaluate \( y \) and \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \)
Now we substitute \( x = \frac{\pi}{4} \):
1. \( \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \)
2. \( \cos 4 \cdot \frac{\pi}{4} = \cos \pi = -1 \)
3. \( \cos 8 \cdot \frac{\pi}{4} = \cos 2\pi = 1 \)

Thus,
\[ y = \frac{1}{\sqrt{2}} \cdot (-1) \cdot 1 = -\frac{1}{\sqrt{2}} \]

Now, we calculate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -\sin \frac{\pi}{4} \cdot (-1) \cdot 1 - 4 \cdot \frac{1}{\sqrt{2}} \cdot \sin \pi \cdot 1 - 8 \cdot \frac{1}{\sqrt{2}} \cdot (-1) \cdot 0 \]
Since \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \) and \( \sin \pi = 0 \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{2}} + 0 + 0 = \frac{1}{\sqrt{2}} \]

### Step 5: Calculate \( \frac{1}{y} \cdot \frac{dy}{dx} \)
Now we compute:
\[ \frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{-\frac{1}{\sqrt{2}}} \cdot \frac{1}{\sqrt{2}} = -\sqrt{2} \cdot \frac{1}{\sqrt{2}} = -1 \]

### Final Answer
Thus, the value of \( \frac{1}{y} \cdot \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is:
\[ \boxed{-1} \]