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The given task asks to compute the derivative of the function \( y = (x^2 + 1)^{3} - x^3 \) at the point \( x = 1 \).

### Step 1: Differentiate the function
We need to apply the chain rule and the power rule to find \( y' \).

The function is:
\[
y = (x^2 + 1)^3 - x^3
\]

First, differentiate \( (x^2 + 1)^3 \) using the chain rule:
\[
\frac{d}{dx} \left( (x^2 + 1)^3 \right) = 3(x^2 + 1)^2 \cdot \frac{d}{dx} (x^2 + 1) = 3(x^2 + 1)^2 \cdot 2x = 6x(x^2 + 1)^2
\]

Now differentiate \( -x^3 \):
\[
\frac{d}{dx} (-x^3) = -3x^2
\]

So, the derivative of the function is:
\[
y' = 6x(x^2 + 1)^2 - 3x^2
\]

### Step 2: Evaluate at \( x = 1 \)
Now substitute \( x = 1 \) into the derivative:
\[
y'(1) = 6(1)(1^2 + 1)^2 - 3(1^2)
\]
\[
y'(1) = 6(1)(1 + 1)^2 - 3(1)
\]
\[
y'(1) = 6(1)(2)^2 - 3 = 6(4) - 3 = 24 - 3 = 21
\]

### Step 3: Solve for the logarithmic terms
From the answer options, it seems the expression \( \ln 2 \) appears. Based on the options given, there may be a logarithmic interpretation of the function or its evaluation. However, the standard derivative yields \( 21 \), which doesn't directly match the given choices.

I recommend reviewing the problem context, especially in relation to the usage of \( \ln 2 \), as this may pertain to a specific method or transformation.

Would you like further details, or do you have any specific questions?

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Here are five related questions that can expand your understanding:
1. How does the chain rule apply when differentiating powers of composite functions?
2. What are the key differences between the product rule and the chain rule?
3. How do you handle logarithmic functions when they appear in differentiation problems?
4. What is the general strategy for differentiating polynomial expressions?
5. How does implicit differentiation work in comparison to explicit differentiation?

**Tip**: Always double-check if logarithmic functions are involved when the answer choices include terms like \( \ln \).

Вычислить производную функции y = (x^2 + 1)^3 - x^3 в заданной точке x = 1.

This problem demonstrates how to differentiate the function y = (x^2 + 1)^3 - x^3 at x = 1 using the chain rule and power rule. It walks through the step-by-step process of solving the derivative and applying the rules of calculus.

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