Find `dy/dx if x= cosy`

Answer

To find \(\frac{dy}{dx}\) given that \(x = \cos y\), we can follow these steps:

### Step 1: Differentiate both sides with respect to \(x\)
We start with the equation:
\[ x = \cos y \]
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(x) = \frac{d}{dx}(\cos y) \]

### Step 2: Apply the differentiation rules
The derivative of \(x\) with respect to \(x\) is \(1\). For the right side, we apply the chain rule:
\[ 1 = -\sin y \cdot \frac{dy}{dx} \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Now, we can isolate \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{1}{\sin y} \]

### Step 4: Rewrite in terms of cosecant
We can express \(-\frac{1}{\sin y}\) in terms of cosecant:
\[ \frac{dy}{dx} = -\csc y \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) when \(x = \cos y\) is:
\[ \frac{dy}{dx} = -\csc y \]

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