Textbook Question
In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
y = (1 / x²), (−1, 1)
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 3.5
Find the derivatives of the functions in Exercises 1–42.
𝔂 = (x + 1)² (x² + 2x)
Verified step by step guidance1
Identify the function as a product of two functions: \( u(x) = (x + 1)^2 \) and \( v(x) = (x^2 + 2x) \).
Apply the product rule for differentiation, which states that if \( y = u(x) \cdot v(x) \), then \( y' = u'(x) \cdot v(x) + u(x) \cdot v'(x) \).
Differentiate \( u(x) = (x + 1)^2 \) using the chain rule: \( u'(x) = 2(x + 1) \cdot 1 = 2(x + 1) \).
Differentiate \( v(x) = x^2 + 2x \) using the power rule: \( v'(x) = 2x + 2 \).
Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the product rule formula to find \( y' \).
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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. Derivatives are fundamental in calculus for understanding rates of change and are denoted as f'(x) or dy/dx.
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05:44
Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as seen in the given function.
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05:18The Product Rule
Chain Rule
The chain rule is a method for differentiating composite functions. If a function y is defined as a composition of two functions, such as y = f(g(x)), the chain rule states that the derivative is dy/dx = f'(g(x)) * g'(x). This rule is crucial when dealing with functions that are nested within each other, which may occur in more complex expressions.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = cos u, u = −x/3
Textbook Question
In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.
f(x) = { 2x − 1, x ≥ 0
x² + 2x + 7, x < 0
Textbook Question
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x⁻¹/² sec (2x)²
Textbook Question
The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is
A(q) = (km / q) + cm + (hq / 2),
where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).
Find dA/dq and d²A/dq².
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.