Use part one of the fundamental theorem of calculus to find the derivative of the function. g(x) = *in(6 + t²) dt 9'(x)=In(6+42) Need Help? Read It
Use part one of the fundamental theorem of calculus to find the derivative of the function. g(x) = *in(6 + t²) dt 9'(x)=In(6+42) Need Help? Read It
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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Use the fundamental theorem of calc to find the derivative of the functions
![**Finding the Derivative Using the Fundamental Theorem of Calculus**
The question involves using part one of the Fundamental Theorem of Calculus to determine the derivative of the given function.
Given:
\[ g(x) = \int_{1}^{x} \ln(6 + t^2) \, dt \]
We are tasked with finding \( g'(x) \).
According to part one of the Fundamental Theorem of Calculus, if \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F'(x) = f(x) \).
Thus, in this case:
\[ g'(x) = \ln(6 + x^2) \]
The image shows an incorrect step:
\[ g'(x) = \ln(6 + 4^2) \]
with a marked error indicated by a red cross.
Here, \( g'(x) \) is mistakenly simplified as \( \ln(6 + 16) \), which is incorrect according to the theorem.
For further help, the image references a "Need Help?" section with a button labeled "Read It".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff77aaeb6-32e9-4475-82f9-455a261f2944%2F2e1cb187-45b0-484b-90b6-8bb1f3e21671%2F9er09wi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Derivative Using the Fundamental Theorem of Calculus**
The question involves using part one of the Fundamental Theorem of Calculus to determine the derivative of the given function.
Given:
\[ g(x) = \int_{1}^{x} \ln(6 + t^2) \, dt \]
We are tasked with finding \( g'(x) \).
According to part one of the Fundamental Theorem of Calculus, if \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F'(x) = f(x) \).
Thus, in this case:
\[ g'(x) = \ln(6 + x^2) \]
The image shows an incorrect step:
\[ g'(x) = \ln(6 + 4^2) \]
with a marked error indicated by a red cross.
Here, \( g'(x) \) is mistakenly simplified as \( \ln(6 + 16) \), which is incorrect according to the theorem.
For further help, the image references a "Need Help?" section with a button labeled "Read It".
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