https://file.aiquickdraw.com/mathbot/file/1727449903324x6nbok06.jpg

The expression in the image is the derivative of a function raised to the power of 10. The function inside the parentheses is:

\[
f(x) = 4x^4 + 2x^3 - x + 1
\]

We need to compute:

\[
\frac{d}{dx} \left( 4x^4 + 2x^3 - x + 1 \right)^{10}
\]

To differentiate this expression, we apply the **chain rule**. The chain rule states:

\[
\frac{d}{dx} \left( [u(x)]^n \right) = n [u(x)]^{n-1} \cdot \frac{d}{dx} u(x)
\]

In this case:
- \( u(x) = 4x^4 + 2x^3 - x + 1 \)
- \( n = 10 \)

Let's go step-by-step:

1. Differentiate the outer function \((u(x))^{10}\):
   \[
   10 \cdot (4x^4 + 2x^3 - x + 1)^9
   \]

2. Now, multiply by the derivative of the inner function \( u(x) = 4x^4 + 2x^3 - x + 1 \):
   \[
   u'(x) = \frac{d}{dx} \left( 4x^4 + 2x^3 - x + 1 \right) = 16x^3 + 6x^2 - 1
   \]

Thus, the derivative of the given expression is:

\[
\frac{d}{dx} \left( 4x^4 + 2x^3 - x + 1 \right)^{10} = 10 \cdot (4x^4 + 2x^3 - x + 1)^9 \cdot (16x^3 + 6x^2 - 1)
\]

This is the complete derivative.

Would you like further clarification on any part of this process or additional details?

Here are some related questions to deepen your understanding:
1. How do you apply the chain rule to a composite function in general?
2. What is the derivative of a polynomial function like \( 4x^4 + 2x^3 - x + 1 \)?
3. How would the derivative change if the power was not 10 but a different number?
4. What happens if one of the terms inside the parentheses is a trigonometric function?
5. How do we use the product rule in combination with the chain rule when necessary?

**Tip:** When applying the chain rule, always differentiate the outer function first, keeping the inner function unchanged, and then multiply by the derivative of the inner function.

Find the derivative of the expression (4x^4 + 2x^3 - x + 1)^10 with respect to x.

Learn how to find the derivative of (4x^4 + 2x^3 - x + 1)^10 using the chain rule in calculus. This problem involves polynomial functions and applying the chain rule for composite functions. Suitable for advanced high school calculus students.

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