`tan (e^x+5)`

Answer

To find the differential coefficient of the function \( f(x) = \tan(e^x + 5) \) with respect to \( x \), we will use the chain rule of differentiation.

### Step-by-Step Solution:

1. **Identify the outer and inner functions**:
- The outer function is \( \tan(u) \) where \( u = e^x + 5 \).
- The inner function is \( u = e^x + 5 \).

2. **Differentiate the outer function**:
- The derivative of \( \tan(u) \) is \( \sec^2(u) \).
- Therefore, \( \frac{d}{du} \tan(u) = \sec^2(u) \).

3. **Differentiate the inner function**:
- The derivative of \( u = e^x + 5 \) is \( \frac{d}{dx}(e^x + 5) = e^x + 0 = e^x \).

4. **Apply the chain rule**:
- According to the chain rule, \( \frac{d}{dx} \tan(u) = \frac{d}{du} \tan(u) \cdot \frac{du}{dx} \).
- Thus, we have:
\[ \frac{d}{dx} \tan(e^x + 5) = \sec^2(e^x + 5) \cdot e^x \]

5. **Final result**:
- Therefore, the derivative of \( f(x) = \tan(e^x + 5) \) is:
\[ f'(x) = e^x \sec^2(e^x + 5) \]

### Summary:
The differential coefficient of \( \tan(e^x + 5) \) with respect to \( x \) is \( e^x \sec^2(e^x + 5) \).