Domanda del libro di testo
Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.
184
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 3.9.24
Find the derivative of the following functions.
y = x In x - x
Guida verificata passo dopo passo1
Identify the function for which you need to find the derivative: \( y = x \ln(x) - x \).
Apply the derivative rules: Use the product rule for \( x \ln(x) \) and the power rule for \( -x \).
For the term \( x \ln(x) \), apply the product rule: \( (u \cdot v)' = u'v + uv' \), where \( u = x \) and \( v = \ln(x) \).
Calculate the derivatives: \( u' = 1 \) and \( v' = \frac{1}{x} \). Substitute these into the product rule: \( (x \ln(x))' = 1 \cdot \ln(x) + x \cdot \frac{1}{x} \).
For the term \( -x \), apply the power rule: \( (x^n)' = nx^{n-1} \). Here, \( n = 1 \), so the derivative is \( -1 \). Combine the results to find the derivative of the entire function.
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:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.
Video consigliato:
Percorso guidato
05:44
Derivatives
Product Rule
The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as the term x ln(x) in the given function.
Video consigliato:
05:18The Product Rule
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately equal to 2.71828. It is a fundamental function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, which is crucial when applying the product rule to differentiate functions involving natural logarithms, such as ln(x) in the given function.
Video consigliato:
05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
63–74. Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = log₈ |tan x|
268
views
Domanda del libro di testo
Find the derivative of the following functions.
y = In |x²-1|
239
views
Domanda del libro di testo
If f is differentiable at a, must f be continuous at a?
453
views
Domanda del libro di testo
Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (sec x) = sec x tan x
363
views
Domanda del libro di testo
75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+x²)^sin x
190
views