Domanda del libro di testo
Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).
352
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 3.4.70
Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).
f(x) = 1/x
Guida verificata passo dopo passo1
Step 1: Identify the function f(x) = \(\frac{1}{x}\). This can be rewritten using exponent notation as f(x) = x^{-1}.
Step 2: Find the first derivative f'(x) by applying the power rule. The power rule states that if f(x) = x^n, then f'(x) = n \(\cdot\) x^{n-1}. For f(x) = x^{-1}, f'(x) = -1 \(\cdot\) x^{-2}.
Step 3: Simplify the expression for the first derivative. f'(x) = -x^{-2} can be rewritten as f'(x) = -\(\frac{1}{x^2}\).
Step 4: Find the second derivative f''(x) by differentiating f'(x) = -x^{-2} again using the power rule. f''(x) = 2 \(\cdot\) x^{-3}.
Step 5: Simplify the expression for the second derivative. f''(x) = 2x^{-3} can be rewritten as f''(x) = \(\frac{2}{x^3}\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
First Derivative
The first derivative of a function, denoted as f'(x), represents the rate of change of the function with respect to its variable. For the function f(x) = 1/x, the first derivative can be found using the power rule or quotient rule, indicating how the function's value changes as x varies.
Video consigliato:
07:09
The First Derivative Test: Finding Local Extrema
Second Derivative
The second derivative, denoted as f''(x), is the derivative of the first derivative. It provides information about the curvature of the function and can indicate concavity. For f(x) = 1/x, calculating the second derivative involves differentiating f'(x) again, revealing how the rate of change itself is changing.
Video consigliato:
06:02The Second Derivative Test: Finding Local Extrema
Higher-Order Derivatives
Higher-order derivatives refer to derivatives taken multiple times. The third derivative, f'''(x), is the derivative of the second derivative and can provide insights into the behavior of the function beyond just its slope and curvature. For f(x) = 1/x, finding the third derivative involves a systematic application of differentiation rules to the second derivative.
Video consigliato:
02:42Higher Order Derivatives
Pratica correlata
Domanda del libro di testo
49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
f (x) = (sin x)^In x; a = π/2
187
views
Domanda del libro di testo
Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).
408
views
Domanda del libro di testo
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
s(t) = 4√t - 1/4t⁴+t+1
305
views
Domanda del libro di testo
Parabolic motion An arrow is shot into the air and moves along the parabolic path y=x(50−x) (see figure). The horizontal component of velocity is always 30 ft/s. What is the vertical component of velocity when (a) x=10 and (b) x=40? <IMAGE>
309
views
Domanda del libro di testo
Find the derivative of the following functions.
y = cot x / (1 + csc x)
405
views