Find `dy/dx if x-7y=siny`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x - 7y = \sin(y)\), we will differentiate both sides of the equation with respect to \(x\). Here’s the step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ x - 7y = \sin(y) \]
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x) - \frac{d}{dx}(7y) = \frac{d}{dx}(\sin(y)) \]

### Step 2: Apply differentiation rules
Using the rules of differentiation:
- The derivative of \(x\) with respect to \(x\) is \(1\).
- The derivative of \(7y\) with respect to \(x\) is \(7 \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\sin(y)\) with respect to \(x\) is \(\cos(y) \frac{dy}{dx}\) (again using the chain rule).

So we have:
\[ 1 - 7 \frac{dy}{dx} = \cos(y) \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Now, we need to collect all terms involving \(\frac{dy}{dx}\) on one side:
\[ 1 = 7 \frac{dy}{dx} + \cos(y) \frac{dy}{dx} \]

### Step 4: Factor out \(\frac{dy}{dx}\)
We can factor out \(\frac{dy}{dx}\) from the right side:
\[ 1 = \frac{dy}{dx} (7 + \cos(y)) \]

### Step 5: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\) by dividing both sides by \((7 + \cos(y))\):
\[ \frac{dy}{dx} = \frac{1}{7 + \cos(y)} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{1}{7 + \cos(y)} \]

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