Use implicit differentiation to find \( \frac{d y}{d x} \). \( x=\sec y \)

To find \( \frac{d y}{d x} \) using implicit differentiation for the equation \( x = \sec y \), we will follow these steps: 1. Differentiate both sides of the equation with respect to \( x \). 2. Use the chain rule for the right side since \( y \) is a function of \( x \). 3. Solve for \( \frac{d y}{d x} \). Let's start by differentiating both sides: Given: \[ x = \sec y \] Differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(x) = \frac{d}{dx}(\sec y) \] The left side becomes: \[ 1 \] For the right side, we apply the chain rule: \[ \frac{d}{dx}(\sec y) = \sec y \tan y \cdot \frac{d y}{d x} \] Putting it all together, we have: \[ 1 = \sec y \tan y \cdot \frac{d y}{d x} \] Now, we can solve for \( \frac{d y}{d x} \): \[ \frac{d y}{d x} = \frac{1}{\sec y \tan y} \] To simplify this expression, we can use the identities \( \sec y = \frac{1}{\cos y} \) and \( \tan y = \frac{\sin y}{\cos y} \): \[ \frac{d y}{d x} = \frac{1}{\frac{1}{\cos y} \cdot \frac{\sin y}{\cos y}} = \frac{\cos^2 y}{\sin y} \] Thus, the final result for \( \frac{d y}{d x} \) is: \[ \frac{d y}{d x} = \frac{\cos^2 y}{\sin y} \]