Domanda del libro di testo
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.2.25b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 1/x+1; a = -1/2;5
Guida verificata passo dopo passo1
Step 1: Identify the function f(x) = \(\frac{1}{x}\) + 1 and the points a = -\(\frac{1}{2}\) and a = 5 where we need to evaluate the derivative f'(a).
Step 2: Find the derivative of the function f(x). The function can be rewritten as f(x) = x^{-1} + 1. Use the power rule for derivatives, which states that \(\frac{d}{dx}\)[x^n] = nx^{n-1}, to find f'(x).
Step 3: Apply the power rule to f(x) = x^{-1} + 1. The derivative of x^{-1} is -x^{-2}, and the derivative of a constant (1) is 0. Therefore, f'(x) = -x^{-2}.
Step 4: Evaluate f'(x) at a = -\(\frac{1}{2}\). Substitute x = -\(\frac{1}{2}\) into f'(x) = -x^{-2} to find f'(-\(\frac{1}{2}\)).
Step 5: Evaluate f'(x) at a = 5. Substitute x = 5 into f'(x) = -x^{-2} to find f'(5).
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Derivatives
A derivative represents the rate of change of a function with respect to its 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 a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Derivatives
Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If f(x) = g(x)/h(x), the derivative f'(x) is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential when dealing with functions that involve division, such as the function f(x) = 1/(x+1) in the given question.
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Evaluating Derivatives at Specific Points
Evaluating the derivative at a specific point involves substituting the value of that point into the derivative function. This process provides the slope of the tangent line to the function at that particular point, which can be useful for understanding the behavior of the function. In this case, we need to compute f'(-1/2) and f'(5) to analyze the function's behavior at these values.
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