Find `dy/dx if 10x-4y=siny`

Answer

To find \(\frac{dy}{dx}\) for the equation \(10x - 4y = \sin y\), we will differentiate both sides of the equation with respect to \(x\).

### Step-by-Step Solution:

1. **Differentiate both sides of the equation:**
\[ \frac{d}{dx}(10x - 4y) = \frac{d}{dx}(\sin y) \]

2. **Apply the differentiation:**
- The derivative of \(10x\) with respect to \(x\) is \(10\).
- The derivative of \(-4y\) with respect to \(x\) is \(-4\frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\sin y\) with respect to \(x\) is \(\cos y \cdot \frac{dy}{dx}\) (again using the chain rule).

So, we have:
\[ 10 - 4\frac{dy}{dx} = \cos y \cdot \frac{dy}{dx} \]

3. **Rearrange the equation to isolate \(\frac{dy}{dx}\):**
\[ 10 = \cos y \cdot \frac{dy}{dx} + 4\frac{dy}{dx} \]
\[ 10 = \left(\cos y + 4\right)\frac{dy}{dx} \]

4. **Solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{10}{\cos y + 4} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{10}{\cos y + 4} \]