Textbook Question
Evaluate and simplify y'.
y = 2x√2
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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 3.R.85b
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
b. d/dx ((f(x) / g(x)) |x=
Verified step by step guidance1
Step 1: Identify the functions f(x) and g(x) from the table provided. Note their values and any derivatives given at specific points.
Step 2: Recall the quotient rule for derivatives, which states that if you have a function h(x) = f(x)/g(x), then the derivative h'(x) is given by: \( h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2} \).
Step 3: Substitute the values of f(x), f'(x), g(x), and g'(x) at the point x = a from the table into the quotient rule formula.
Step 4: Simplify the expression obtained in Step 3 by performing the necessary arithmetic operations.
Step 5: Ensure that the denominator (g(x))^2 is not zero at x = a to avoid division by zero, and verify the final expression for correctness.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. Understanding how to compute derivatives is essential for analyzing the behavior of functions.
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Derivatives
Quotient Rule
The Quotient Rule is a formula used to find the derivative of a function that is the ratio of two other functions. Specifically, if you have a function h(x) = f(x) / g(x), the derivative h'(x) is given by (g(x)f'(x) - f(x)g'(x)) / (g(x))^2. This rule is crucial when dealing with derivatives of fractions.
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Evaluating Derivatives at a Point
Evaluating a derivative at a specific point involves substituting the value of the variable into the derivative function. This process provides the slope of the tangent line to the function at that particular point, which is important for understanding the function's behavior in that vicinity.
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Related Practice
Textbook Question
9–61. Evaluate and simplify y'.
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Textbook Question
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Textbook Question
9–61. Evaluate and simplify y'.
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Textbook Question
The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Compute the average rate of population growth from 1950 to 1960.
Textbook Question
Use differentiation to verify each equation.
d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).