Textbook Question
Find the derivatives of the functions in Exercises 1–42.
______
𝓻 = √2θ sinθ
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 3.27
Find the derivatives of the functions in Exercises 1–42.
𝔂 = x² sin² (2x²)
Verified step by step guidance1
Step 1: Identify the function y = x² sin²(2x²). Notice that this is a product of two functions: u(x) = x² and v(x) = sin²(2x²).
Step 2: Apply the product rule for derivatives, which states that if y = u(x) * v(x), then y' = u'(x) * v(x) + u(x) * v'(x).
Step 3: Differentiate u(x) = x². The derivative u'(x) is 2x.
Step 4: Differentiate v(x) = sin²(2x²). Use the chain rule: v'(x) = 2 * sin(2x²) * cos(2x²) * (d/dx)(2x²).
Step 5: Calculate (d/dx)(2x²), which is 4x. Substitute back into the expression for v'(x) to get v'(x) = 2 * sin(2x²) * cos(2x²) * 4x.
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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 a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.
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Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, as seen in the given function y = x² sin²(2x²).
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Chain Rule
The chain rule is a technique for differentiating composite functions. If a function y is defined as a composition of two functions, say y = f(g(x)), the chain rule states that the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is particularly useful when dealing with functions that include nested expressions, such as sin²(2x²) in the given problem.
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Related Practice
Textbook Question
Using the Alternative Formula for Derivatives
Use the formula
f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)
to find the derivative of the functions in Exercises 23–26.
g(x) = 1 + √x
Textbook Question
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
g(x) = 8 / x², (2, 2)
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
f(x) = √(7 + x sec x)
Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
2y³/² + xy − x = 0
Textbook Question
In Exercises 9–18, write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x.
y = (4 − 3x)⁹