Textbook Question
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
xy + 2x - 5y = 2, (3, 2)
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.43
Implicit Differentiation
In Exercises 43–50, find by implicit differentiation.
xy + 2x + 3y = 1
Verified step by step guidance1
Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use the chain rule.
Differentiate the term xy. Use the product rule: if u = x and v = y, then the derivative of uv is u'v + uv'. Here, u' = 1 and v' = dy/dx.
Differentiate the term 2x. The derivative of 2x with respect to x is simply 2.
Differentiate the term 3y. Since y is a function of x, use the chain rule: the derivative is 3(dy/dx).
Combine all the differentiated terms and set them equal to the derivative of the constant on the right side of the equation, which is 0. Solve for dy/dx to find the derivative of y with respect to x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule to account for the dependent variable's implicit relationship.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to terms involving the dependent variable, treating it as a function of the independent variable. This means that when differentiating a term like y, we multiply by dy/dx, the derivative of y with respect to x.
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05:02Intro to the Chain Rule
Solving for dy/dx
After applying implicit differentiation to an equation, the next step is to isolate dy/dx to find the derivative of y with respect to x. This involves rearranging the differentiated equation to express dy/dx in terms of x and y. This process is crucial for understanding how y changes in relation to x, especially in contexts where y cannot be easily expressed as a function of x.
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5:02Solving Logarithmic Equations
Related Practice
Textbook Question
In Exercises 43–50, find by implicit differentiation.
__
√xy = 1
Textbook Question
The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.
Textbook Question
In Exercises 41–58, find dy/dt.
y = (t⁻³/⁴ sin(t))⁴/³
Textbook Question
Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).
Textbook Question
Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?
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