Find the slope of the line tangent to h(x) = f(x)g(x) at x = 6, given that the line tangent to the graph of f(x) at x = 6 is y = 2x - 1, and the line tangent to the graph of g(x) at x = 6 is y = 13 - 3x.
Find the slope of the line tangent to h(x) = f(x)g(x) at x = 6, given that the line tangent to the graph of f(x) at x = 6 is y = 2x - 1, and the line tangent to the graph of g(x) at x = 6 is y = 13 - 3x.
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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How to find derivative of tangent line of two equations see image below
![**Problem Statement:**
Find the slope of the line tangent to \( h(x) = f(x)g(x) \) at \( x = 6 \), given that the line tangent to the graph of \( f(x) \) at \( x = 6 \) is \( y = 2x - 1 \), and the line tangent to the graph of \( g(x) \) at \( x = 6 \) is \( y = 13 - 3x \).
**Solution:**
To find the slope of the tangent to \( h(x) = f(x)g(x) \), we use the product rule in differentiation:
\[
h'(x) = f'(x)g(x) + f(x)g'(x)
\]
Given:
- The slope of the tangent to \( f(x) \) at \( x = 6 \) is 2, since the line is \( y = 2x - 1 \).
- The slope of the tangent to \( g(x) \) at \( x = 6 \) is -3, since the line is \( y = 13 - 3x \).
Therefore:
- \( f'(6) = 2 \)
- \( g'(6) = -3 \)
Now substitute these values into the product rule expression:
\[
h'(6) = f'(6)g(6) + f(6)g'(6)
\]
Assuming initial conditions for \( f(6) \) and \( g(6) \) are known (or given in a complete problem context), calculate \( h'(6) \) using the given derivatives.
**Note:** This transcription assumes a standard approach to solving such a problem in calculus. The actual values for \( f(6) \) and \( g(6) \) are necessary to find \( h'(6) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F181283bb-83dc-4f09-86f1-fac04e6d4723%2F53421cd1-bb47-438c-ae8f-573847dd9942%2Fcr0toof_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the slope of the line tangent to \( h(x) = f(x)g(x) \) at \( x = 6 \), given that the line tangent to the graph of \( f(x) \) at \( x = 6 \) is \( y = 2x - 1 \), and the line tangent to the graph of \( g(x) \) at \( x = 6 \) is \( y = 13 - 3x \).
**Solution:**
To find the slope of the tangent to \( h(x) = f(x)g(x) \), we use the product rule in differentiation:
\[
h'(x) = f'(x)g(x) + f(x)g'(x)
\]
Given:
- The slope of the tangent to \( f(x) \) at \( x = 6 \) is 2, since the line is \( y = 2x - 1 \).
- The slope of the tangent to \( g(x) \) at \( x = 6 \) is -3, since the line is \( y = 13 - 3x \).
Therefore:
- \( f'(6) = 2 \)
- \( g'(6) = -3 \)
Now substitute these values into the product rule expression:
\[
h'(6) = f'(6)g(6) + f(6)g'(6)
\]
Assuming initial conditions for \( f(6) \) and \( g(6) \) are known (or given in a complete problem context), calculate \( h'(6) \) using the given derivatives.
**Note:** This transcription assumes a standard approach to solving such a problem in calculus. The actual values for \( f(6) \) and \( g(6) \) are necessary to find \( h'(6) \).
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