https://file.aiquickdraw.com/mathbot/file/1726627791681jm8ioszs.jpg

The derivatives of the given function \( f(x) = x^4 + 3x^3 - 4x^{-1} \) are:

\[
f'(x) = 4x^3 + 9x^2 + \frac{4}{x^2}
\]

\[
f''(x) = 12x^2 + 18x - \frac{8}{x^3}
\]

\[
f'''(x) = 24x + 18 + \frac{24}{x^4}
\]

\[
f^{(4)}(x) = 24 - \frac{96}{x^5}
\]

Would you like any further details or have questions on these derivatives?

Here are five related questions to explore:
1. What are the critical points of the function \( f(x) \)?
2. How can the behavior of the function be analyzed using its second derivative?
3. What is the concavity of the function based on the second derivative?
4. How does the fourth derivative help in analyzing the function's higher-order behavior?
5. Can you find any inflection points for this function?

**Tip:** When calculating derivatives, always simplify expressions to avoid algebraic errors and make further analysis easier.

Find f', f'', f''', and f^(4) for the function f(x) = x^4 + 3x^3 - 4x^-1.

Find the first, second, third, and fourth derivatives of the function f(x) = x^4 + 3x^3 - 4x^-1. This calculus problem involves using the power rule for derivatives to compute f'(x), f''(x), f'''(x), and f^(4)(x). Suitable for Grades 11-12 or first-year university students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation