Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x³/3 + x²/2 + x/4
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 3.3.4
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
w = 3z⁷ − 7z³ + 21z²
Verified step by step guidance1
Step 1: Identify the function for which you need to find the derivatives. Here, the function is \( w = 3z^7 - 7z^3 + 21z^2 \).
Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that the derivative of \( z^n \) is \( nz^{n-1} \).
Step 3: Calculate the first derivative \( \frac{dw}{dz} \) by applying the power rule: \( \frac{d}{dz}(3z^7) = 21z^6 \), \( \frac{d}{dz}(-7z^3) = -21z^2 \), and \( \frac{d}{dz}(21z^2) = 42z \). Combine these results to get the first derivative.
Step 4: To find the second derivative, apply the power rule again to the first derivative. Differentiate each term of the first derivative.
Step 5: Calculate the second derivative \( \frac{d^2w}{dz^2} \) by applying the power rule to each term of the first derivative obtained in Step 3. Combine these results to get the second derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The first derivative of a function provides information about its slope, while the second derivative gives insight into the function's concavity and acceleration.
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Derivatives
Power Rule
The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of ax^n, where a is a constant and n is a real number. According to this rule, the derivative is calculated by multiplying the coefficient by the exponent and then reducing the exponent by one. This rule simplifies the process of finding derivatives for polynomial functions.
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Higher-Order Derivatives
Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative gives the rate of change, the second derivative provides information about the curvature of the graph, and further derivatives can indicate more complex behaviors of the function. Understanding higher-order derivatives is essential for analyzing the behavior of functions in greater detail.
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Related Practice
Textbook Question
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
Textbook Question
Rates of Change
Speed of a rocket At t sec after liftoff, the height of a rocket is 3t² ft. How fast is the rocket climbing 10 sec after liftoff?
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴
Textbook Question
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
Textbook Question
Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
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f(x) = x² + 2x, x₀ = 1, dx = 0.1