$$ \left. \begin{array} { l } { y = ( 1 + x ^ { 2 } ) ^ { 3 } } \ { \frac { d y } { d x } = } \end{array} ight. $$

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$$ \left. \begin{array} { l } { y = ( 1 + x ^ { 2 } ) ^ { 3 } } \ { \frac { d y } { d x } = } \end{array} ight. $$

UpStudy AI · Accepted Answer

To find the derivative of $$ y = (1 + x^2)^3 $$ with respect to $$ x $$, we can use the **chain rule** from calculus. The chain rule is essential when dealing with composite functions, where one function is nested inside another.

### Step-by-Step Solution:

1. **Identify the Outer and Inner Functions:**
   
   - **Outer function:** $$ u^3 $$, where $$ u = 1 + x^2 $$
   - **Inner function:** $$ u = 1 + x^2 $$

2. **Differentiate the Outer Function with Respect to $$ u $$:**
   
   $$
   \frac{d}{du}(u^3) = 3u^2
   $$

3. **Differentiate the Inner Function with Respect to $$ x $$:**
   
   $$
   \frac{du}{dx} = \frac{d}{dx}(1 + x^2) = 2x
   $$

4. **Apply the Chain Rule:**
   
   The chain rule states that:
   
   $$
   \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
   $$
   
   Substituting the derivatives we found:
   
   $$
   \frac{dy}{dx} = 3u^2 \cdot 2x = 6x \cdot u^2
   $$
   
   Remembering that $$ u = 1 + x^2 $$, we substitute back:
   
   $$
   \frac{dy}{dx} = 6x(1 + x^2)^2
   $$

### Final Answer:

$$
\frac{dy}{dx} = 6x(1 + x^2)^2
$$

This derivative represents the rate at which $$ y $$ changes with respect to $$ x $$ for the given function.
The derivative of $$ y = (1 + x^2)^3 $$ with respect to $$ x $$ is $$ \frac{dy}{dx} = 6x(1 + x^2)^2 $$.

\( \left. \begin{array} { l } { y = ( 1 + x ^ { 2 } ) ^ { 3 } } \\ { \frac { d y } { d x } = } \end{array} \right. \)

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