Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:**Problem Statement:**
Compute \(\frac{d^2}{dx^2} \left( -x^4 - 3x^2 - x \right)\).
**Context:**
The task is to find the second derivative of the polynomial function \(-x^4 - 3x^2 - x\) with respect to \(x\). This involves differentiating the given function twice to determine how the rate of change itself changes with \(x\).
**No Graphs or Diagrams Present.**
Explanation of Solution:
1. **First Derivative** (\(f'(x)\)):
- Start by differentiating each term of the function with respect to \(x\).
- \(-x^4 \) differentiates to \(-4x^3\).
- \(-3x^2 \) differentiates to \(-6x\).
- \(-x\) differentiates to \(-1\).
2. **Second Derivative** (\(f''(x)\)):
- Differentiate each term of the first derivative.
- \(-4x^3\) differentiates to \(-12x^2\).
- \(-6x\) differentiates to \(-6\).
- The derivative of \(-1\) is \(0\).
Thus, the second derivative of the function is \(f''(x) = -12x^2 - 6\).
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