derivative of `tan (e^x+5)`

Answer

To find the derivative of the function \( f(x) = \tan(e^x + 5) \), we will use the chain rule. Here are the steps to solve the problem:

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

### Step 2: Differentiate the outer function
The derivative of \( \tan(u) \) with respect to \( u \) is:
\[ \frac{d}{du}(\tan(u)) = \sec^2(u) \]

### Step 3: Differentiate the inner function
Now, we need to find the derivative of the inner function \( u = e^x + 5 \):
\[ \frac{du}{dx} = \frac{d}{dx}(e^x) + \frac{d}{dx}(5) \]
Since the derivative of \( e^x \) is \( e^x \) and the derivative of a constant (5) is 0, we have:
\[ \frac{du}{dx} = e^x + 0 = e^x \]

### Step 4: Apply the chain rule
Now, we can apply the chain rule:
\[ \frac{df}{dx} = \frac{d}{du}(\tan(u)) \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{df}{dx} = \sec^2(u) \cdot e^x \]

### Step 5: Substitute back for \( u \)
Now, substituting back \( u = e^x + 5 \):
\[ \frac{df}{dx} = \sec^2(e^x + 5) \cdot e^x \]

### Final Result
Thus, the derivative of \( \tan(e^x + 5) \) is:
\[ \frac{df}{dx} = e^x \sec^2(e^x + 5) \]

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