https://file.aiquickdraw.com/mathbot/file/1727407872568xciicusj.jpg

The function provided is:

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

To find the derivative \( \frac{dy}{dx} \), we'll need to use the **chain rule** and the **product rule** in the following steps.

### Step 1: Apply the Chain Rule
The outermost part of the function is \( [u]^7 \), where \( u = x + (x + \sin^2(x))^6 \). The derivative of \( u^7 \) with respect to \( u \) is:

\[
\frac{d}{dx}\left( u^7 \right) = 7u^6 \cdot \frac{du}{dx}
\]

### Step 2: Find \( \frac{du}{dx} \)
Now, we need to differentiate \( u = x + (x + \sin^2(x))^6 \). This will also require the chain rule. First, split \( u \) into two parts:

1. The derivative of \( x \) with respect to \( x \) is 1.
2. The second term is \( (x + \sin^2(x))^6 \), so we need to apply the chain rule again.

Let \( v = x + \sin^2(x) \), so:

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

Now, find \( \frac{dv}{dx} \):

\[
\frac{dv}{dx} = \frac{d}{dx} (x + \sin^2(x)) = 1 + 2\sin(x)\cos(x) = 1 + \sin(2x)
\]

Thus,

\[
\frac{du}{dx} = 1 + 6(x + \sin^2(x))^5 \cdot (1 + \sin(2x))
\]

### Step 3: Multiply and Simplify
Finally, we can substitute this expression for \( \frac{du}{dx} \) into the chain rule result from step 1:

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

This is the derivative of the given function.

---

Would you like to see a step-by-step solution of any particular part or need further clarification?

### Related Questions:
1. What are the key concepts in applying the chain rule?
2. How do you differentiate functions with powers, like \( (x^n) \)?
3. Can you explain how the sine and cosine functions are derived?
4. What is the significance of the derivative in a real-world context?
5. How would you compute the second derivative of this function?

### Tip:
When solving complex derivatives, always break down the problem into smaller parts and apply the chain rule step-by-step.

Find the derivative of the function y = [x + (x + sin^2(x))^6]^7

Explore how to differentiate the function y = [x + (x + sin^2(x))^6]^7 using the chain rule and product rule. This problem involves calculus concepts, such as trigonometric functions and multi-layered differentiation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation