Textbook Question
Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?
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 3.3.2
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x² + x + 8
Verified step by step guidance1
Step 1: Identify the function for which you need to find the derivatives. In this case, the function is y = x² + x + 8.
Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that if y = xⁿ, then the derivative dy/dx = n*x^(n-1).
Step 3: Apply the power rule to the first term x². The derivative of x² is 2*x^(2-1) = 2x.
Step 4: Apply the power rule to the second term x. The derivative of x is 1*x^(1-1) = 1.
Step 5: The constant term 8 has a derivative of 0, as the derivative of any constant is zero. Combine the derivatives of each term to find the first derivative: dy/dx = 2x + 1. Then, apply the same process to find the second derivative by differentiating dy/dx = 2x + 1 again.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at any given point.
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Derivatives
Power Rule
The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if a function is in the form of f(x) = x^n, where n is a real number, then its derivative f'(x) is given by f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for functions involving powers of x.
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5:50Power Rules
Second Derivative
The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function's graph and can indicate whether the function is concave up or concave down. The second derivative is useful for analyzing the acceleration of a function and identifying points of inflection where the function changes concavity.
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Related Practice
Textbook Question
In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?
54. x³ + y² = sin²y
Textbook Question
Slopes and Tangent Lines
In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.
Textbook Question
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
Textbook Question
In Exercises 29 and 30, find the slope of the curve at the given points.
(x² + y²)² = (x – y)² at (1,0) and (1,–1)
Textbook Question
Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².