Differentiate `logsinx` with respect to `sqrt(cosx)` .

Explanation

To differentiate \( \log(\sin x) \) with respect to \( \sqrt{\cos x} \), we will use the chain rule and the properties of logarithmic differentiation. Let's break it down step by step.

### Step 1: Define the Functions
Let:
- \( u = \log(\sin x) \)
- \( v = \sqrt{\cos x} \)

We need to find \( \frac{du}{dv} \).
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