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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.3.64
Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,
d/dx (x⁻ᵐ) = −mx⁻ᵐ⁻¹
where m is a positive integer.
Verified step by step guidance1
Start by expressing the function x^(-m) as a fraction: x^(-m) = 1/x^m.
Apply the Quotient Rule for derivatives, which states that if you have a function in the form of u/v, the derivative is (v * du/dx - u * dv/dx) / v^2.
Set u = 1 and v = x^m. Then, compute the derivatives: du/dx = 0 and dv/dx = m * x^(m-1).
Substitute these into the Quotient Rule formula: (x^m * 0 - 1 * m * x^(m-1)) / (x^m)^2.
Simplify the expression: -m * x^(m-1) / x^(2m) = -m * x^(-m-1), which proves the Power Rule for negative integers.
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative Quotient Rule
The Quotient Rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. If you have a function f(x) = g(x)/h(x), the derivative f'(x) is given by (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential for differentiating functions expressed as fractions, such as x⁻ᵐ, which can be rewritten as 1/xᵐ.
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The Quotient Rule
Power Rule for Derivatives
The Power Rule is a basic rule in calculus for finding the derivative of a function of the form f(x) = xⁿ, where n is any real number. The derivative is given by f'(x) = nxⁿ⁻¹. This rule simplifies the process of differentiation and is fundamental for understanding how to handle powers of x, including negative powers.
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5:50Power Rules
Negative Integer Exponents
Negative integer exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, x⁻ᵐ is equivalent to 1/xᵐ. Understanding this concept is crucial when applying the Power Rule to negative exponents, as it involves rewriting the expression in a form suitable for differentiation using the Quotient Rule.
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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
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = (2√x)/(3(1 + √x))
Textbook Question
In Exercises 19–22, find the slope of the curve at the point indicated.
y = x³ − 2x + 7, x = −2
Textbook Question
[Technology Exercise]
Graph the curves in Exercises 39–48.
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.
y = 4x²/⁵ − 2x
Textbook Question
One-Sided Derivatives
Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.