To find the derivative of the function $y = \left(x^2 + 6\right)^2 \left(7x^2 - 2\right)^3$, we can use the product rule in conjunction with the chain rule.

Given: $y = u \cdot v$ where $u = \left(x^2 + 6\right)^2$ and $v = \left(7x^2 - 2\right)^3$.

Step 1: Differentiate $u$ and $v$ separately

1. Differentiate $u = \left(x^2 + 6\right)^2$: $u' = 2 \cdot \left(x^2 + 6\right) \cdot (2x) = 4x \left(x^2 + 6\right)$

2. Differentiate $v = \left(7x^2 - 2\right)^3$: $v' = 3 \cdot \left(7x^2 - 2\right)^2 \cdot (14x) = 42x \left(7x^2 - 2\right)^2$

Step 2: Apply the Product Rule

The product rule states that: $y' = u' \cdot v + u \cdot v'$

Substitute $u$, $u'$, $v$, and $v'$ into this formula: $y' = \left(4x \left(x^2 + 6\right)\right) \cdot \left(7x^2 - 2\right)^3 + \left(x^2 + 6\right)^2 \cdot \left(42x \left(7x^2 - 2\right)^2\right)$

This expression is the derivative of the function.

Would you like a further simplification or have any specific questions?

Related Questions

1. How would you apply the chain rule if there were an additional function inside the parentheses?
2. Can you explain how the product rule works in general?
3. What is the importance of the power rule in finding derivatives of exponential functions?
4. How would the derivative change if the exponents in $u$ or $v$ were different?
5. How would you find the second derivative of this function?

Tip

Always simplify your derivatives as much as possible for clearer expressions and easier computation in subsequent steps.