Find `dy/dx` if :
`ax+by^(2)=cosy`

Answer

To find \( \frac{dy}{dx} \) for the equation \( ax + by^2 = \cos y \), we will use implicit differentiation. Here is the step-by-step solution:

### Step 1: Differentiate both sides with respect to \( x \)
We start with the equation:
\[ ax + by^2 = \cos y \]
Differentiating both sides with respect to \( x \):
\[ \frac{d}{dx}(ax) + \frac{d}{dx}(by^2) = \frac{d}{dx}(\cos y) \]

### Step 2: Apply the differentiation rules
Using the product rule and chain rule:
- The derivative of \( ax \) with respect to \( x \) is \( a \).
- The derivative of \( by^2 \) with respect to \( x \) is \( 2by \frac{dy}{dx} \) (using the chain rule).
- The derivative of \( \cos y \) with respect to \( x \) is \( -\sin y \frac{dy}{dx} \) (using the chain rule).

Putting these together, we have:
\[ a + 2by \frac{dy}{dx} = -\sin y \frac{dy}{dx} \]

### Step 3: Rearranging the equation
Now, we will rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ 2by \frac{dy}{dx} + \sin y \frac{dy}{dx} = -a \]
Factoring out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx}(2by + \sin y) = -a \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{-a}{2by + \sin y} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{-a}{2by + \sin y} \]