Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √2x+1; a= 4
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 3.4.11a
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
Verified step by step guidance1
Identify the function f(w) = \(\frac{w^3 - w}{w}\) and simplify it by dividing each term in the numerator by the denominator.
Simplify the expression: \(\frac{w^3}{w}\) - \(\frac{w}{w}\) = w^2 - 1.
Recognize that the simplified function f(w) = w^2 - 1 is a polynomial function.
Differentiate the simplified function using the power rule: \(\frac{d}{dw}\)(w^2) = 2w and \(\frac{d}{dw}\)(-1) = 0.
Combine the derivatives to find the derivative of the function: f'(w) = 2w.
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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(w) and v(w), the derivative of their product is given by 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 functions that are expressed as the ratio of two other functions. If f(w) = u(w)/v(w), the derivative is given by (u'v - uv')/v². This rule is particularly useful when dealing with fractions in calculus, ensuring that the differentiation accounts for both the numerator and denominator.
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06:43The Quotient Rule
Simplification of Derivatives
Simplification of derivatives involves reducing the expression obtained after applying differentiation rules to its simplest form. This may include factoring, canceling common terms, or combining like terms. Simplifying the result is crucial for clarity and ease of interpretation, especially when further analysis or evaluation is required.
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05:44Derivatives
Related Practice
Textbook Question
Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>
a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4
Textbook Question
62–65. {Use of Tech} Graphing f and f'
a. Graph f with a graphing utility.
f(x)=e^−x tan^−1 x on [0,∞)
Textbook Question
Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
a. (f^-1)'(4)
Textbook Question
City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.
a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?
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