If `y=log(sinx-cosx)`, then `(dy)/(dx)` at `x=(pi)/(2)` is :

Options

A. 0
B. 1
C. `(1)/(2)`
D. `-2`

Answer

To solve the problem \( y = \log(\sin x - \cos x) \) and find \( \frac{dy}{dx} \) at \( x = \frac{\pi}{2} \), we will follow these steps:

### Step 1: Differentiate \( y \)
We start with the function:
\[ y = \log(\sin x - \cos x) \]
To differentiate \( y \) with respect to \( x \), we use the chain rule. The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).

### Step 2: Identify \( u \) and find \( \frac{du}{dx} \)
Here, let:
\[ u = \sin x - \cos x \]
Now, we differentiate \( u \):
\[ \frac{du}{dx} = \cos x + \sin x \]
(The derivative of \( \sin x \) is \( \cos x \) and the derivative of \( -\cos x \) is \( \sin x \).)

### Step 3: Apply the chain rule
Now, we can write:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{\sin x - \cos x} \cdot (\cos x + \sin x) \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{\cos x + \sin x}{\sin x - \cos x} \]

### Step 4: Evaluate \( \frac{dy}{dx} \) at \( x = \frac{\pi}{2} \)
Next, we substitute \( x = \frac{\pi}{2} \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \frac{\cos\left(\frac{\pi}{2}\right) + \sin\left(\frac{\pi}{2}\right)}{\sin\left(\frac{\pi}{2}\right) - \cos\left(\frac{\pi}{2}\right)} \]
Calculating the values:
- \( \cos\left(\frac{\pi}{2}\right) = 0 \)
- \( \sin\left(\frac{\pi}{2}\right) = 1 \)

Substituting these values:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{2}} = \frac{0 + 1}{1 - 0} = \frac{1}{1} = 1 \]

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