Find the derivative of `y = 4^(6x)`.

Answer

To find the derivative of the function \( y = 4^{6x} \), we can follow these steps:

### Step 1: Rewrite the function
The function is given as:
\[ y = 4^{6x} \]
We can express \( 4 \) as \( 2^2 \):
\[ y = (2^2)^{6x} = 2^{12x} \]

### Step 2: Differentiate using the chain rule
Now, we will differentiate \( y = 2^{12x} \) with respect to \( x \). The derivative of \( a^{u} \) (where \( a \) is a constant and \( u \) is a function of \( x \)) is given by:
\[ \frac{dy}{dx} = a^{u} \cdot \ln(a) \cdot \frac{du}{dx} \]
In our case, \( a = 2 \) and \( u = 12x \).

### Step 3: Calculate \( \frac{du}{dx} \)
The derivative of \( u = 12x \) with respect to \( x \) is:
\[ \frac{du}{dx} = 12 \]

### Step 4: Apply the derivative formula
Now, substituting into the derivative formula:
\[ \frac{dy}{dx} = 2^{12x} \cdot \ln(2) \cdot 12 \]

### Step 5: Simplify the expression
Thus, we can simplify the expression:
\[ \frac{dy}{dx} = 12 \cdot 2^{12x} \cdot \ln(2) \]

### Final Answer
The derivative of \( y = 4^{6x} \) is:
\[ \frac{dy}{dx} = 12 \cdot 4^{6x} \cdot \ln(4) \]