If `y=cos(x+y)," then "(dy)/(dx)=`

Options

A. `(cos(x+y))/(1+sin(x+y))`
B. `(sin(x+y))/(1-sin(x+y))`
C. `(-sin(x+y))/(1+sin(x+y))`
D. `(1+(dy)/(dx))sin(x+y)`

Answer

To solve the equation \( y = \cos(x + y) \) and find \( \frac{dy}{dx} \), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides with respect to \( x \)

Starting with the equation:
\[ y = \cos(x + y) \]

We differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}[\cos(x + y)] \]

### Step 2: Apply the chain rule on the right side

Using the chain rule, we differentiate \( \cos(x + y) \):
\[ \frac{d}{dx}[\cos(x + y)] = -\sin(x + y) \cdot \frac{d}{dx}(x + y) \]

### Step 3: Differentiate \( x + y \)

Now, differentiate \( x + y \):
\[ \frac{d}{dx}(x + y) = \frac{d}{dx}(x) + \frac{d}{dx}(y) = 1 + \frac{dy}{dx} \]

### Step 4: Substitute back into the equation

Substituting back into our differentiation result, we have:
\[ \frac{dy}{dx} = -\sin(x + y)(1 + \frac{dy}{dx}) \]

### Step 5: Distribute the right side

Distributing \( -\sin(x + y) \) gives:
\[ \frac{dy}{dx} = -\sin(x + y) - \sin(x + y) \cdot \frac{dy}{dx} \]

### Step 6: Collect all \( \frac{dy}{dx} \) terms on one side

Rearranging the equation to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} + \sin(x + y) \cdot \frac{dy}{dx} = -\sin(x + y) \]

Factoring out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx}(1 + \sin(x + y)) = -\sin(x + y) \]

### Step 7: Solve for \( \frac{dy}{dx} \)

Finally, divide both sides by \( 1 + \sin(x + y) \):
\[ \frac{dy}{dx} = \frac{-\sin(x + y)}{1 + \sin(x + y)} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{-\sin(x + y)}{1 + \sin(x + y)} \]

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