Differentiate the following w.r.t. x :
`1/3e^(x)-5e`

Answer

To differentiate the function \( y = \frac{1}{3} e^x - 5e \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function
We have the function:
\[ y = \frac{1}{3} e^x - 5e \]
Here, \( \frac{1}{3} e^x \) is a term involving \( x \) and \( -5e \) is a constant term (since \( e \) is a constant).

### Step 2: Differentiate each term
To differentiate \( y \) with respect to \( x \), we will differentiate each term separately.

1. **Differentiate \( \frac{1}{3} e^x \)**:
- The derivative of \( e^x \) with respect to \( x \) is \( e^x \).
- Therefore, the derivative of \( \frac{1}{3} e^x \) is:
\[ \frac{d}{dx}\left(\frac{1}{3} e^x\right) = \frac{1}{3} e^x \]

2. **Differentiate \( -5e \)**:
- Since \( -5e \) is a constant, its derivative is:
\[ \frac{d}{dx}(-5e) = 0 \]

### Step 3: Combine the derivatives
Now, we combine the derivatives of both terms:
\[ \frac{dy}{dx} = \frac{1}{3} e^x + 0 = \frac{1}{3} e^x \]

### Final Answer
Thus, the derivative of the function \( y = \frac{1}{3} e^x - 5e \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{3} e^x \]

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