give answer in 2 step with explanation at the end of each step and final answer at the end:ind the derivative of the function:
4
y=G +) -8)

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:ind the derivative of the function:
4
y=G +) -8)

VIPSolutions · Accepted Answer

## Step 1: Identify the Rule to Use

The function is:
$$ y = (3x + 1)^7 \cdot (x^4 - 6)^4 $$

This is a product of two functions, so we use the **product rule**:
$$
\frac{d}{dx}[u \cdot v] = u'v + uv'
$$
Let $ u = (3x+1)^7 $ and $ v = (x^4-6)^4 $.

**Explanation:**  
We need to differentiate each part and then apply the product rule.

---

## Step 2: Differentiate Each Part

Differentiate $ u $ and $ v $:

- $ u' = 7(3x+1)^6 \cdot 3 = 21(3x+1)^6 $
- $ v' = 4(x^4-6)^3 \cdot 4x^3 = 16x^3(x^4-6)^3 $

Now, apply the product rule:
$$
y' = u'v + uv'
$$
$$
y' = 21(3x+1)^6 \cdot (x^4-6)^4 + (3x+1)^7 \cdot 16x^3 (x^4-6)^3
$$

**Explanation:**  
We used the chain rule for each derivative and then substituted into the product rule formula.

---

## Final Answer

$$
\boxed{
y' = 21(3x+1)^6 (x^4-6)^4 + 16x^3 (3x+1)^7 (x^4-6)^3
}
$$

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give answer in 2 step with explanation at the end of each step and final answer at the end:ind the derivative of the function: 4 y=G +) -8)

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