`(5+7x)^(6)`

Answer

To differentiate the function \( y = (5 + 7x)^6 \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
Let:
- Outer function: \( u^6 \) where \( u = 5 + 7x \)
- Inner function: \( u = 5 + 7x \)

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( u^n \) is \( n \cdot u^{n-1} \):
\[ \frac{dy}{du} = 6u^5 \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( u = 5 + 7x \):
\[ \frac{du}{dx} = 0 + 7 = 7 \]

### Step 4: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 6(5 + 7x)^5 \cdot 7 \]

### Step 5: Simplify the expression
Now we can simplify:
\[ \frac{dy}{dx} = 42(5 + 7x)^5 \]

Thus, the derivative of \( y = (5 + 7x)^6 \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 42(5 + 7x)^5 \]

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