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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![**Find the indicated derivative.**
Given \( f(x) = x^3 \), we need to find \(\left. \frac{df}{dx} \right|_{x = -11} \).
The attempted solution provided in the image is:
\[
\left. \frac{df}{dx} \right|_{x = -11} = \frac{-8}{x^{\frac{3}{2}}} + \frac{10}{x^{\frac{1}{2}}}
\]
However, this solution is incorrect, as indicated by the red "X" symbol next to it.
**Explanation of the Provided Solution Steps:**
- The provided solution attempts to evaluate the derivative by showing fractional exponents in terms of \( x \).
- The terms \(\frac{-8}{x^{3/2}}\) and \(\frac{10}{x^{1/2}}\) do not relate correctly to the derivative of the function \( f(x) = x^3 \).
Let's correctly solve the problem:
1. **Find the derivative of \( f(x) \):**
\[
f(x) = x^3
\]
\[
\frac{df}{dx} = 3x^2
\]
2. **Evaluate the derivative at \( x = -11 \):**
\[
\left. \frac{df}{dx} \right|_{x = -11} = 3(-11)^2 = 3 \cdot 121 = 363
\]
So, the correct value of the derivative at \( x = -11 \) is \(\boxed{363}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7e0cd55c-6f84-412a-ac81-9a79eaea810f%2F3d61ba7a-bb52-42d8-afad-b460e52b7f39%2F2wcfxp_processed.png&w=3840&q=75)
Transcribed Image Text:**Find the indicated derivative.**
Given \( f(x) = x^3 \), we need to find \(\left. \frac{df}{dx} \right|_{x = -11} \).
The attempted solution provided in the image is:
\[
\left. \frac{df}{dx} \right|_{x = -11} = \frac{-8}{x^{\frac{3}{2}}} + \frac{10}{x^{\frac{1}{2}}}
\]
However, this solution is incorrect, as indicated by the red "X" symbol next to it.
**Explanation of the Provided Solution Steps:**
- The provided solution attempts to evaluate the derivative by showing fractional exponents in terms of \( x \).
- The terms \(\frac{-8}{x^{3/2}}\) and \(\frac{10}{x^{1/2}}\) do not relate correctly to the derivative of the function \( f(x) = x^3 \).
Let's correctly solve the problem:
1. **Find the derivative of \( f(x) \):**
\[
f(x) = x^3
\]
\[
\frac{df}{dx} = 3x^2
\]
2. **Evaluate the derivative at \( x = -11 \):**
\[
\left. \frac{df}{dx} \right|_{x = -11} = 3(-11)^2 = 3 \cdot 121 = 363
\]
So, the correct value of the derivative at \( x = -11 \) is \(\boxed{363}\).
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