Find `dy/dx if y=cos(logx)`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = \cos(\log x)\), we will use the chain rule of differentiation. Let's go through the steps:

### Step 1: Identify the outer and inner functions
In this case, we have:
- Outer function: \(y = \cos(\theta)\) where \(\theta = \log x\)
- Inner function: \(\theta = \log x\)

### Step 2: Differentiate the outer function
The derivative of \(y = \cos(\theta)\) with respect to \(\theta\) is:
\[ \frac{dy}{d\theta} = -\sin(\theta) \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \(\theta = \log x\) with respect to \(x\):
\[ \frac{d\theta}{dx} = \frac{1}{x} \]

### Step 4: Apply the chain rule
According to the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{d\theta}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = -\sin(\log x) \cdot \frac{1}{x} \]

### Step 5: Simplify the expression
Thus, we can write:
\[ \frac{dy}{dx} = -\frac{\sin(\log x)}{x} \]

### Final Answer
The derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{\sin(\log x)}{x} \]

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