**Title: Finding the Line Tangent to Combined Curves****Introduction:**Understanding how to determine the line tangent to different types of curves at a specific point is an essential concept in calculus. In this lesson, we will explore how to find the line tangents to combined curves at a given point.**Problem Statement:**Suppose the line tangent to the graph of \( f \) at \( x = 4 \) is \( y = 2x + 1 \), and the line tangent to the graph of \( g \) at \( x = 4 \) is \( y = 5x - 3 \). We need to find the line tangent to the following curves at \( x = 4 \):a. \( y = f(x)g(x) \)b. \( y = \frac{f(x)}{g(x)} \)**Formulas and Calculations:**To solve this problem, we need to apply the derivative rules for product and quotient functions. Let's denote \( f(4) \) and \( g(4) \) as the values of the functions at \( x = 4 \).1. The derivative of a product \( y = f(x)g(x) \):\[ y' = f'(x)g(x) + f(x)g'(x) \]2. The derivative of a quotient \( y = \frac{f(x)}{g(x)} \):\[ y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]Given:- \( f'(4) = 2 \) (slope from the tangent line \( y = 2x + 1 \))- \( g'(4) = 5 \) (slope from the tangent line \( y = 5x - 3 \))We apply the above rules at \( x = 4 \).**Results:**- The equation of the line tangent to the curve \( y = f(x)g(x) \) at \( x = 4 \) is:\[ y = [\frac{d}{dx} (f(x)g(x))]_{x=4} \]- The equation of the line tangent to the curve \( y = \frac{f(x)}{g(x)} \) at \( x = 4 \) is:

Name: **Title: Finding the Line Tangent to Combined Curves****Introduction:
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Title: Finding the Line Tangent to Combined Curves****Introduction:**Understanding how to determine the line tangent to different types of curves at a specific point is an essential concept in calculus. In this lesson, we will explore how to find