Textbook Question
Parallel tangent lines Find the two points where the curve x² + xy + y² = 7 crosses the x-axis, and show that the tangent lines to the curve at these points are parallel. What is the common slope of these tangent lines?
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 3.9.20
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = (2√x)/(3(1 + √x))
Verified step by step guidance1
Step 1: Begin by identifying the function y = \(\frac{2\sqrt{x}\)}{3(1 + \(\sqrt{x}\))}. This is a rational function where both the numerator and the denominator involve square roots.
Step 2: Apply the quotient rule for derivatives, which states that if you have a function y = \(\frac{u(x)}{v(x)}\), then the derivative dy/dx is given by \(\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, u(x) = 2\(\sqrt{x}\) and v(x) = 3(1 + \(\sqrt{x}\)).
Step 3: Find the derivative of the numerator u(x) = 2\(\sqrt{x}\). The derivative u'(x) can be found using the chain rule. Recall that \(\sqrt{x}\) = x^{1/2}, so u'(x) = 2 \(\cdot\) \(\frac{1}{2}\)x^{-1/2} = x^{-1/2}.
Step 4: Find the derivative of the denominator v(x) = 3(1 + \(\sqrt{x}\)). The derivative v'(x) involves differentiating 1 + \(\sqrt{x}\). The derivative of \(\sqrt{x}\) is \(\frac{1}{2}\)x^{-1/2}, so v'(x) = 3 \(\cdot\) \(\frac{1}{2}\)x^{-1/2} = \(\frac{3}{2}\)x^{-1/2}.
Step 5: Substitute u(x), u'(x), v(x), and v'(x) into the quotient rule formula: dy/dx = \(\frac{x^{-1/2}\) \(\cdot\) 3(1 + \(\sqrt{x}\)) - 2\(\sqrt{x}\) \(\cdot\) \(\frac{3}{2}\)x^{-1/2}}{(3(1 + \(\sqrt{x}\)))^2}. Simplify the expression to find the derivative dy/dx.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Derivatives
Differential Form
Differential form refers to the expression of derivatives in terms of differentials, typically denoted as dy and dx. In this context, dy represents the change in the function y as x changes, and is calculated using the derivative. This form is particularly useful for understanding how small changes in x affect y.
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Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are functions of x, the derivative is given by dy/dx = (v(du/dx) - u(dv/dx)) / v². This rule is essential for differentiating functions like y = (2√x)/(3(1 + √x)), where both the numerator and denominator are functions of x.
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Related Practice
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.
__
x + √xy = 6, (4, 1)
Textbook Question
Finding Derivative Functions and Values
Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.
f(x) = 4 – x²; f′(−3), f′(0), f′(1)
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Textbook Question
In Exercises 19–22, find the slope of the curve at the point indicated.
y = x³ − 2x + 7, x = −2
Textbook Question
The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s
b. volume?
Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
xy² − 4x³/² − y = 0