If `y=cos x cos 2x cos 4x cos 8x`, then `(dy)/(dx)" at "x=(pi)/(2)` is

Options

A. 1
B. 0
C. 4
D. 16

Answer

To find the derivative \(\frac{dy}{dx}\) of the function \(y = \cos x \cos 2x \cos 4x \cos 8x\) at \(x = \frac{\pi}{2}\), we can follow these steps:

### Step 1: Rewrite the Function
We start with the given function:
\[ y = \cos x \cos 2x \cos 4x \cos 8x \]

### Step 2: Use Trigonometric Identities
We can use the identity \(2 \sin x \cos x = \sin 2x\) to simplify the expression. We will multiply and divide by \(2 \sin x\):
\[ y = \frac{\sin 2x}{2 \sin x} \cdot \cos 2x \cos 4x \cos 8x \]

### Step 3: Simplify Further
Continuing to apply the identity, we can rewrite the function:
\[ y = \frac{\sin 4x}{4 \sin x} \cdot \cos 4x \cos 8x \]
and then,
\[ y = \frac{\sin 8x}{8 \sin x} \cdot \cos 8x \]
and finally,
\[ y = \frac{\sin 16x}{16 \sin x} \]

### Step 4: Differentiate the Function
Now we differentiate \(y\) using the quotient rule:
\[ \frac{dy}{dx} = \frac{(16 \cos 16x)(\sin x) - (\sin 16x)(\cos x)}{(16 \sin x)^2} \]

### Step 5: Evaluate at \(x = \frac{\pi}{2}\)
Now we substitute \(x = \frac{\pi}{2}\):
- \(\sin\left(\frac{\pi}{2}\right) = 1\)
- \(\cos\left(\frac{\pi}{2}\right) = 0\)
- \(\sin(16 \cdot \frac{\pi}{2}) = \sin(8\pi) = 0\)
- \(\cos(16 \cdot \frac{\pi}{2}) = \cos(8\pi) = 1\)

Substituting these values into the derivative:
\[ \frac{dy}{dx} = \frac{(16 \cdot 1)(1) - (0)(0)}{(16 \cdot 1)^2} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{16}{256} = \frac{1}{16} \]

### Final Result
Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{2}\) is:
\[ \frac{dy}{dx} = 1 \]