`(3-4x)^(5)`

Answer

To differentiate the function \( y = (3 - 4x)^5 \) with respect to \( x \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
Let:
- Outer function: \( u^5 \) where \( u = 3 - 4x \)
- Inner function: \( u = 3 - 4x \)

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( u^n \) is \( n \cdot u^{n-1} \). Therefore, the derivative of \( u^5 \) is:
\[ \frac{dy}{du} = 5u^4 \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( u = 3 - 4x \):
\[ \frac{du}{dx} = 0 - 4 = -4 \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 5(3 - 4x)^4 \cdot (-4) \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ \frac{dy}{dx} = -20(3 - 4x)^4 \]

Thus, the derivative of \( y = (3 - 4x)^5 \) is:
\[ \frac{dy}{dx} = -20(3 - 4x)^4 \]