If `y=log(cose^(x)),"then find "dy/dx`.

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \log(\cos(e^x)) \), we will use the chain rule and the properties of logarithmic and trigonometric functions. Here’s a step-by-step solution:

### Step 1: Differentiate \( y \)
Given:
\[ y = \log(\cos(e^x)) \]

We will differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx} \left( \log(\cos(e^x)) \right) \]

### Step 2: Apply the Chain Rule
Using the chain rule for differentiation, we have:
\[ \frac{dy}{dx} = \frac{1}{\cos(e^x)} \cdot \frac{d}{dx}(\cos(e^x)) \]

### Step 3: Differentiate \( \cos(e^x) \)
Now, we need to differentiate \( \cos(e^x) \):
\[ \frac{d}{dx}(\cos(e^x)) = -\sin(e^x) \cdot \frac{d}{dx}(e^x) \]

Since the derivative of \( e^x \) is \( e^x \), we have:
\[ \frac{d}{dx}(\cos(e^x)) = -\sin(e^x) \cdot e^x \]

### Step 4: Substitute Back
Now we substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\cos(e^x)} \cdot \left(-\sin(e^x) \cdot e^x\right) \]

### Step 5: Simplify
This simplifies to:
\[ \frac{dy}{dx} = -\frac{e^x \sin(e^x)}{\cos(e^x)} \]

### Step 6: Use Trigonometric Identity
Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), we can rewrite this as:
\[ \frac{dy}{dx} = -e^x \tan(e^x) \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -e^x \tan(e^x) \]

---