Find the derivative of `y = 2^(3x)`

Answer

To find the derivative of the function \( y = 2^{3x} \), we can follow these steps:

### Step 1: Identify the function
The given function is:
\[ y = 2^{3x} \]

### Step 2: Use the chain rule for differentiation
To differentiate \( y = 2^{3x} \), we can use the formula for the derivative of an exponential function \( a^{u} \), which is given by:
\[ \frac{d}{dx}(a^u) = a^u \cdot \ln(a) \cdot \frac{du}{dx} \]
where \( a \) is a constant and \( u \) is a function of \( x \).

In our case, \( a = 2 \) and \( u = 3x \).

### Step 3: Differentiate \( u = 3x \)
The derivative of \( u = 3x \) with respect to \( x \) is:
\[ \frac{du}{dx} = 3 \]

### Step 4: Apply the differentiation formula
Now we can apply the differentiation formula:
\[ \frac{dy}{dx} = 2^{3x} \cdot \ln(2) \cdot \frac{du}{dx} \]
Substituting \( \frac{du}{dx} \):
\[ \frac{dy}{dx} = 2^{3x} \cdot \ln(2) \cdot 3 \]

### Step 5: Simplify the expression
We can rearrange the expression:
\[ \frac{dy}{dx} = 3 \cdot 2^{3x} \cdot \ln(2) \]

### Final Answer
Thus, the derivative of \( y = 2^{3x} \) is:
\[ \frac{dy}{dx} = 3 \cdot 2^{3x} \cdot \ln(2) \]

---