If `siny+e^(-xcosy)=e`, then `(dy)/(dx)` at `(1,pi)`, is

Options

A. `siny`
B. `-xcosy`
C. e
D. `siny-xcosy`

Answer

To find \(\frac{dy}{dx}\) for the equation \( \sin y + e^{-x \cos y} = e \) at the point \((1, \pi)\), we will follow these steps:

### Step 1: Differentiate both sides of the equation
We start with the given equation:
\[ \sin y + e^{-x \cos y} = e \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}(\sin y) + \frac{d}{dx}(e^{-x \cos y}) = \frac{d}{dx}(e) \]
Since \(e\) is a constant, its derivative is \(0\):
\[ \frac{d}{dx}(\sin y) + \frac{d}{dx}(e^{-x \cos y}) = 0 \]

### Step 2: Apply the chain rule and product rule
Using the chain rule on \(\sin y\):
\[ \cos y \frac{dy}{dx} \]
For the term \(e^{-x \cos y}\), we apply the chain rule and product rule:
\[ \frac{d}{dx}(e^{-x \cos y}) = e^{-x \cos y} \cdot \frac{d}{dx}(-x \cos y) \]
Now, applying the product rule to \(-x \cos y\):
\[ \frac{d}{dx}(-x \cos y) = -\cos y - x \frac{d}{dx}(\cos y) = -\cos y + x \sin y \frac{dy}{dx} \]
Thus, we have:
\[ \frac{d}{dx}(e^{-x \cos y}) = e^{-x \cos y} \left(-\cos y + x \sin y \frac{dy}{dx}\right) \]

### Step 3: Combine the derivatives
Now substituting back into our differentiated equation:
\[ \cos y \frac{dy}{dx} + e^{-x \cos y} \left(-\cos y + x \sin y \frac{dy}{dx}\right) = 0 \]
Expanding this gives:
\[ \cos y \frac{dy}{dx} - e^{-x \cos y} \cos y + e^{-x \cos y} x \sin y \frac{dy}{dx} = 0 \]

### Step 4: Factor out \(\frac{dy}{dx}\)
Rearranging the equation:
\[ \left(\cos y + e^{-x \cos y} x \sin y\right) \frac{dy}{dx} = e^{-x \cos y} \cos y \]
Thus,
\[ \frac{dy}{dx} = \frac{e^{-x \cos y} \cos y}{\cos y + e^{-x \cos y} x \sin y} \]

### Step 5: Substitute the point \((1, \pi)\)
Now we substitute \(x = 1\) and \(y = \pi\):
- \(\cos \pi = -1\)
- \(\sin \pi = 0\)
- \(e^{-1 \cdot (-1)} = e^1 = e\)

Substituting these values into our equation:
\[ \frac{dy}{dx} = \frac{e \cdot (-1)}{-1 + e \cdot 1 \cdot 0} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{-e}{-1} = e \]

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