Find the derivative of the following w.r.t. x :
`ax+by^(2)=cosy`

Answer

To find the derivative of the equation \( ax + by^2 = \cos y \) with respect to \( x \), we will use implicit differentiation. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ ax + by^2 = \cos y \]

2. **Differentiate both sides with respect to \( x \)**:
- The left-hand side:
- Differentiate \( ax \):
\[ \frac{d}{dx}(ax) = a \]
- Differentiate \( by^2 \) using the chain rule:
\[ \frac{d}{dx}(by^2) = b \cdot 2y \cdot \frac{dy}{dx} = 2by \frac{dy}{dx} \]
- The right-hand side:
- Differentiate \( \cos y \) using the chain rule:
\[ \frac{d}{dx}(\cos y) = -\sin y \cdot \frac{dy}{dx} \]

3. **Combine the derivatives**:
\[ a + 2by \frac{dy}{dx} = -\sin y \frac{dy}{dx} \]

4. **Rearrange the equation to isolate \( \frac{dy}{dx} \)**:
- Move all terms involving \( \frac{dy}{dx} \) to one side:
\[ 2by \frac{dy}{dx} + \sin y \frac{dy}{dx} = -a \]
- Factor out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx}(2by + \sin y) = -a \]

5. **Solve for \( \frac{dy}{dx} \)**:
\[ \frac{dy}{dx} = \frac{-a}{2by + \sin y} \]

### Final Result:
\[ \frac{dy}{dx} = \frac{-a}{2by + \sin y} \]