Find the slope of the tangent line to the curve below at the point (5, 3). √x + 4y + /5xy 12.783359663462 V slope = Submit Question DELL O Sign out 9:08
Find the slope of the tangent line to the curve below at the point (5, 3). √x + 4y + /5xy 12.783359663462 V slope = Submit Question DELL O Sign out 9:08
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...
Related questions
Question
![### Calculating the Slope of the Tangent Line
In this section, you'll learn how to find the slope of the tangent line to a given curve at a specific point. Below is an example problem, which involves implicit differentiation to find the slope of the tangent line.
#### Example Problem:
**Given Curve:**
\[ \sqrt{x} + 4y + \sqrt{5xy} = 12.783359663462 \]
**Point of Tangency:**
\[ (5, 3) \]
We begin by applying implicit differentiation to the equation of the curve with respect to \( x \). This involves differentiating each term of the equation while treating \( y \) as a function of \( x \) (i.e., using the chain rule for terms involving \( y \)).
Here's a step by step outline:
1. **Differentiate each term of the given equation implicitly with respect to \( x \)**:
- For \(\sqrt{x}\), use the chain rule:
\[ \frac{d}{dx}\left(\sqrt{x}\right) = \frac{1}{2\sqrt{x}} \]
- For \(4y\), apply the chain rule considering \( y \) as a function of \( x \):
\[ \frac{d}{dx}\left(4y\right) = 4\frac{dy}{dx} \]
- For \(\sqrt{5xy}\), use the product rule and chain rule:
\[ \frac{d}{dx}\left(\sqrt{5xy}\right) = \frac{1}{2\sqrt{5xy}} \cdot \left(5y + 5x\frac{dy}{dx}\right) = \frac{5y + 5x\frac{dy}{dx}}{2\sqrt{5xy}} \]
- For the constant value (12.783359663462), its derivative is 0.
2. **Set up the implicit differentiation equation**:
\[
\frac{1}{2\sqrt{x}} + 4\frac{dy}{dx} + \frac{5y + 5x\frac{dy}{dx}}{2\sqrt{5xy}} = 0
\]
3. **Substitute the given point \((5, 3)\) into the differentiated equation and solve](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf909276-6ad3-4144-b98c-235c5a32e436%2Fd073259d-85d4-4d2a-92b3-8402144bca60%2Fnparuy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Slope of the Tangent Line
In this section, you'll learn how to find the slope of the tangent line to a given curve at a specific point. Below is an example problem, which involves implicit differentiation to find the slope of the tangent line.
#### Example Problem:
**Given Curve:**
\[ \sqrt{x} + 4y + \sqrt{5xy} = 12.783359663462 \]
**Point of Tangency:**
\[ (5, 3) \]
We begin by applying implicit differentiation to the equation of the curve with respect to \( x \). This involves differentiating each term of the equation while treating \( y \) as a function of \( x \) (i.e., using the chain rule for terms involving \( y \)).
Here's a step by step outline:
1. **Differentiate each term of the given equation implicitly with respect to \( x \)**:
- For \(\sqrt{x}\), use the chain rule:
\[ \frac{d}{dx}\left(\sqrt{x}\right) = \frac{1}{2\sqrt{x}} \]
- For \(4y\), apply the chain rule considering \( y \) as a function of \( x \):
\[ \frac{d}{dx}\left(4y\right) = 4\frac{dy}{dx} \]
- For \(\sqrt{5xy}\), use the product rule and chain rule:
\[ \frac{d}{dx}\left(\sqrt{5xy}\right) = \frac{1}{2\sqrt{5xy}} \cdot \left(5y + 5x\frac{dy}{dx}\right) = \frac{5y + 5x\frac{dy}{dx}}{2\sqrt{5xy}} \]
- For the constant value (12.783359663462), its derivative is 0.
2. **Set up the implicit differentiation equation**:
\[
\frac{1}{2\sqrt{x}} + 4\frac{dy}{dx} + \frac{5y + 5x\frac{dy}{dx}}{2\sqrt{5xy}} = 0
\]
3. **Substitute the given point \((5, 3)\) into the differentiated equation and solve
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning