evaluate the derivative of the following functions by the extended power rule (x^2)^2

To evaluate the derivative of the function \((x^2)^2\) using the extended power rule, let's go through the process step by step.

### Step 1: Simplify the Function
The given function is \((x^2)^2\). First, simplify it:
\[
(x^2)^2 = x^{4}
\]

### Step 2: Apply the Power Rule
The power rule for differentiation is:
\[
\frac{d}{dx}(x^n) = n x^{n-1}
\]
Here, \(n = 4\). Applying this rule to \(x^4\):

\[
\frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3
\]

### Final Answer:
The derivative of \((x^2)^2\) is:
\[
4x^3
\]

Would you like further clarification on any step?

### Related Questions:
1. How would you differentiate \((x^3)^2\) using the extended power rule?
2. Can you apply the chain rule to \((x^2)^2\)?
3. What happens if we differentiate \((x^2 + 1)^2\)?
4. How do we differentiate a function of the form \((x^n)^m\)?
5. How would you compute the second derivative of \(x^4\)?

### Tip:
Always simplify functions when possible before differentiating—it makes the process quicker and reduces errors!

Explore how to find the derivative of the function (x^2)^2 using the extended power rule. Simplify the function and apply the power rule step by step to find the solution. Suitable for advanced high school calculus students.

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