The function you're asked to differentiate is:

$y = \left[x + \left(x + \sin^2(x)\right)^6 \right]^7$

Steps to Differentiate:

1. Use the Chain Rule: Since the outermost part is raised to the 7th power, apply the chain rule. Let $u = x + (x + \sin^2(x))^6$, and find $\frac{dy}{dx}$ using:

   $\frac{dy}{dx} = 7 \left[u\right]^6 \cdot \frac{du}{dx}$

2. Differentiate $u = x + (x + \sin^2(x))^6$: Now, let's differentiate $u$ with respect to $x$. Again, use the chain rule on $(x + \sin^2(x))^6$.

   $\frac{du}{dx} = \frac{d}{dx} \left[x + (x + \sin^2(x))^6\right]$

   • The derivative of $x$ is 1.
   • Now, for $(x + \sin^2(x))^6$, let $v = x + \sin^2(x)$, so:

   $\frac{d}{dx} \left[(v)^6\right] = 6(v)^5 \cdot \frac{dv}{dx}$

3. Differentiate $v = x + \sin^2(x)$: $\frac{dv}{dx} = \frac{d}{dx} \left[x + \sin^2(x)\right]$

   • The derivative of $x$ is 1.
   • The derivative of $\sin^2(x)$ is $2\sin(x)\cos(x) = \sin(2x)$ (using the chain rule and trigonometric identity).

   So, $\frac{dv}{dx} = 1 + \sin(2x)$

4. Putting it all together:

   Now, combining all these pieces:

   $\frac{du}{dx} = 1 + 6\left(x + \sin^2(x)\right)^5 \cdot \left(1 + \sin(2x)\right)$

5. Final derivative:

   $\frac{dy}{dx} = 7\left(x + (x + \sin^2(x))^6\right)^6 \cdot \left[1 + 6\left(x + \sin^2(x)\right)^5 \cdot (1 + \sin(2x))\right]$

This is the derivative of the function.

Let me know if you'd like further clarifications or a step-by-step breakdown of any part of the process!

Here are five related questions for practice:

1. How would you apply the product rule along with the chain rule?
2. What are some common trigonometric identities used in differentiation?
3. How does the chain rule change if there are more layers of nested functions?
4. What is the derivative of $\sin^2(x)$ without using identities?
5. How does implicit differentiation work with trigonometric functions?

Tip: The chain rule is especially powerful for functions raised to powers, and it’s important to differentiate each nested function carefully.