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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![To solve this problem, follow these steps:
We need to find the derivative of the function:
\[ 5e^{5x^4 + 3x^6} \]
using the chain rule.
The chain rule states that the derivative of a composed function \( h(x) = f(g(x)) \) is given by:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
In this case:
- The outer function \( f(g) = 5e^g \)
- The inner function \( g(x) = 5x^4 + 3x^6 \)
### Steps:
1. Differentiate the outer function while keeping the inner function intact:
- The derivative of \( 5e^g \) with respect to \( g \) is \( 5e^g \)
2. Differentiate the inner function \( g(x) = 5x^4 + 3x^6 \):
\[ g'(x) = 20x^3 + 18x^5 \]
3. Multiply the results from steps 1 and 2:
\[ \text{The derivative} = 5e^{5x^4 + 3x^6} \cdot (20x^3 + 18x^5) \]
4. Simplify:
\[ = 5e^{5x^4 + 3x^6} \cdot 20x^3 + 5e^{5x^4 + 3x^6} \cdot 18x^5 \]
\[ = 100x^3 e^{5x^4 + 3x^6} + 90x^5 e^{5x^4 + 3x^6} \]
Thus, the derivative of \( 5e^{5x^4 + 3x^6} \) is \( 100x^3 e^{5x^4 + 3x^6} + 90x^5 e^{5x^4 + 3x^6} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee642632-5a94-4462-8244-d32551abf7cd%2Ffd18004e-3bcc-4f7a-a0b6-4495a6bbb90b%2Fagh4j7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To solve this problem, follow these steps:
We need to find the derivative of the function:
\[ 5e^{5x^4 + 3x^6} \]
using the chain rule.
The chain rule states that the derivative of a composed function \( h(x) = f(g(x)) \) is given by:
\[ h'(x) = f'(g(x)) \cdot g'(x) \]
In this case:
- The outer function \( f(g) = 5e^g \)
- The inner function \( g(x) = 5x^4 + 3x^6 \)
### Steps:
1. Differentiate the outer function while keeping the inner function intact:
- The derivative of \( 5e^g \) with respect to \( g \) is \( 5e^g \)
2. Differentiate the inner function \( g(x) = 5x^4 + 3x^6 \):
\[ g'(x) = 20x^3 + 18x^5 \]
3. Multiply the results from steps 1 and 2:
\[ \text{The derivative} = 5e^{5x^4 + 3x^6} \cdot (20x^3 + 18x^5) \]
4. Simplify:
\[ = 5e^{5x^4 + 3x^6} \cdot 20x^3 + 5e^{5x^4 + 3x^6} \cdot 18x^5 \]
\[ = 100x^3 e^{5x^4 + 3x^6} + 90x^5 e^{5x^4 + 3x^6} \]
Thus, the derivative of \( 5e^{5x^4 + 3x^6} \) is \( 100x^3 e^{5x^4 + 3x^6} + 90x^5 e^{5x^4 + 3x^6} \).
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