Find `dy/dx if 3x-6y=siny`

Answer

To find \(\frac{dy}{dx}\) for the equation \(3x - 6y = \sin y\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 3x - 6y = \sin y \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the differentiation rules
- The derivative of \(3x\) with respect to \(x\) is \(3\).
- The derivative of \(-6y\) with respect to \(x\) is \(-6 \frac{dy}{dx}\) (using the chain rule, since \(y\) is a function of \(x\)).
- The derivative of \(\sin y\) with respect to \(x\) is \(\cos y \frac{dy}{dx}\) (again using the chain rule).

Putting this all together, we have:
\[ 3 - 6 \frac{dy}{dx} = \cos y \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Now, we want to isolate \(\frac{dy}{dx}\). We can rearrange the equation:
\[ 3 = \cos y \frac{dy}{dx} + 6 \frac{dy}{dx} \]
This can be factored as:
\[ 3 = \frac{dy}{dx} (\cos y + 6) \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\) by dividing both sides by \((\cos y + 6)\):
\[ \frac{dy}{dx} = \frac{3}{\cos y + 6} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{3}{\cos y + 6} \]

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