Find the derivative of the following w.r.t. x :
`e^(cosx)`

Answer

To find the derivative of \( y = e^{\cos x} \) with respect to \( x \), we can follow these steps:

### Step 1: Write the function
We start with the function:
\[ y = e^{\cos x} \]

### Step 2: Take the natural logarithm of both sides
Taking the natural logarithm of both sides gives us:
\[ \ln y = \ln(e^{\cos x}) \]

### Step 3: Apply the logarithmic property
Using the property of logarithms, \( \ln(a^b) = b \ln a \), we can simplify the right side:
\[ \ln y = \cos x \cdot \ln e \]
Since \( \ln e = 1 \), this simplifies to:
\[ \ln y = \cos x \]

### Step 4: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(\cos x) \]

### Step 5: Apply the chain rule
Using the chain rule on the left side, we have:
\[ \frac{1}{y} \frac{dy}{dx} = -\sin x \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now, we multiply both sides by \( y \) to solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = -y \sin x \]

### Step 7: Substitute back for \( y \)
Substituting back \( y = e^{\cos x} \):
\[ \frac{dy}{dx} = -e^{\cos x} \sin x \]

### Final Answer
Thus, the derivative of \( e^{\cos x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -e^{\cos x} \sin x \]

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