Questions: Find the derivative. [ fracdd xleft[2 xleft(x^7+1right)^6right] ]

$Find the derivative. [ fracdd xleft[2 xleft(x^7+1right)^6right] ]$

Transcript text: Find the derivative. \[ \frac{d}{d x}\left[2 x\left(x^{7}+1\right)^{6}\right] \]

Solution

Solution Steps

To find the derivative of the given function, we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions is given by the derivative of the first function times the second function plus the first function times the derivative of the second function. The chain rule is used to differentiate composite functions.

Step 1: Define the Function

We start with the function: \[ f(x) = 2x(x^7 + 1)^6 \]

Step 2: Apply the Product Rule

The product rule states: \[ \frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v' \] Here, let: \[ u = 2x \quad \text{and} \quad v = (x^7 + 1)^6 \]

Step 3: Differentiate $u$ and $v$

First, we find the derivatives of $u$ and $v$: \[ u' = \frac{d}{dx}[2x] = 2 \] \[ v' = \frac{d}{dx}[(x^7 + 1)^6] \]

Step 4: Apply the Chain Rule to $v'$

Using the chain rule for $v$: \[ v' = 6(x^7 + 1)^5 \cdot \frac{d}{dx}[x^7 + 1] = 6(x^7 + 1)^5 \cdot 7x^6 = 42x^6(x^7 + 1)^5 \]

Step 5: Combine Using the Product Rule

Now, combine the results using the product rule: \[ \frac{d}{dx}[2x(x^7 + 1)^6] = 2 \cdot (x^7 + 1)^6 + 2x \cdot 42x^6(x^7 + 1)^5 \]

Step 6: Simplify the Expression

Simplify the expression: \[ 2(x^7 + 1)^6 + 84x^7(x^7 + 1)^5 \]