Textbook Question
At all times, the length of a rectangle is twice the width w of the rectangleas the area of the rectangle changes with respect to time t.
a. Find an equation relating A to w.
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 7a
Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
f(x) = x(x-1)
Verified step by step guidance1
Step 1: Identify the function f(x) = x(x-1) as a product of two functions, u(x) = x and v(x) = x-1.
Step 2: Recall the Product Rule for derivatives, which states that if you have two functions u(x) and v(x), the derivative of their product is given by (uv)' = u'v + uv'.
Step 3: Differentiate u(x) = x to find u'(x). The derivative of x with respect to x is 1, so u'(x) = 1.
Step 4: Differentiate v(x) = x-1 to find v'(x). The derivative of x is 1 and the derivative of a constant is 0, so v'(x) = 1.
Step 5: Apply the Product Rule: f'(x) = u'(x)v(x) + u(x)v'(x). Substitute u(x), v(x), u'(x), and v'(x) into this formula to find the derivative of f(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule
The Product Rule is a formula used to find the derivative of the product of two functions. If you have two functions, u(x) and v(x), the derivative of their product is given by f'(x) = u'v + uv'. This rule is essential when differentiating expressions where two functions are multiplied together, allowing for a systematic approach to finding the derivative.
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The Product Rule
Quotient Rule
The Quotient Rule is used to differentiate a function that is the ratio of two other functions. If f(x) = u(x)/v(x), the derivative is given by f'(x) = (u'v - uv')/v^2. This rule is particularly useful when dealing with fractions of functions, ensuring that both the numerator and denominator are appropriately accounted for in the differentiation process.
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Simplification of Derivatives
Simplification of derivatives involves reducing the expression obtained after differentiation to its simplest form. This may include factoring, combining like terms, or canceling common factors. Simplifying the result is crucial for clarity and ease of interpretation, especially when further analysis or evaluation of the derivative is required.
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Related Practice
Textbook Question
The volume V of a sphere of radius r changes over time t.
c. At what rate is the radius changing if the volume increases at 10 in³ when the radius is 5 inches?
Textbook Question
The volume V of a sphere of radius r changes over time t.
b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?
Textbook Question
Use differentiation to verify each equation.
d/dx(x / √1−x²) = 1 / (1−x²)^3/2.
Textbook Question
An equation of the line tangent to the graph of f at the point (2,7) is y = 4x−1. Find f(2) and f′(2).
Textbook Question
An equation of the line tangent to the graph of g at x = 3 is y = 5x + 4. Find g(3) and g′(3).