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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I am stuck on this derivative.
![### Calculating the Derivative of a Function
**Function Definition:**
\[ K(r) = e^r + r e \]
**Derivative Calculation:**
To find \( K'(r) \), we need to differentiate each term of the function \( K(r) \) with respect to \( r \).
\[ K' = \frac{d}{dr} ( e^r ) + \frac{d}{dr} ( r e ) \]
**First Term:**
The derivative of \( e^r \) with respect to \( r \):
\[ \frac{d}{dr} ( e^r ) = e^r \]
**Second Term:**
The derivative of \( r e \) with respect to \( r \):
The term \( r e \) is highlighted and it seems there is a request for help with computing this derivative. Let's break it down:
- Recognize that \( e \) is a constant with respect to \( r \).
- The derivative of \( r \) (a linear term) is \( 1 \).
\[ \frac{d}{dr} ( r e ) = e \]
Therefore, summing the derivatives of each term:
\[ K'(r) = e^r + e \]
**Note:**
There was a handwritten note requesting help with the highlighted derivative part \( \frac{d}{dr} (r e) \). From the context, it's clear that understanding constants when differentiating is key here. Since \( e \) is a constant, the differential simplifies straightforwardly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F2c93714c-ea54-4814-ab09-8ccf79830215%2Fyp3kcqh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Derivative of a Function
**Function Definition:**
\[ K(r) = e^r + r e \]
**Derivative Calculation:**
To find \( K'(r) \), we need to differentiate each term of the function \( K(r) \) with respect to \( r \).
\[ K' = \frac{d}{dr} ( e^r ) + \frac{d}{dr} ( r e ) \]
**First Term:**
The derivative of \( e^r \) with respect to \( r \):
\[ \frac{d}{dr} ( e^r ) = e^r \]
**Second Term:**
The derivative of \( r e \) with respect to \( r \):
The term \( r e \) is highlighted and it seems there is a request for help with computing this derivative. Let's break it down:
- Recognize that \( e \) is a constant with respect to \( r \).
- The derivative of \( r \) (a linear term) is \( 1 \).
\[ \frac{d}{dr} ( r e ) = e \]
Therefore, summing the derivatives of each term:
\[ K'(r) = e^r + e \]
**Note:**
There was a handwritten note requesting help with the highlighted derivative part \( \frac{d}{dr} (r e) \). From the context, it's clear that understanding constants when differentiating is key here. Since \( e \) is a constant, the differential simplifies straightforwardly.
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