Domanda del libro di testo
Define the acceleration of an object moving in a straight line.
577
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 3.4.19
Derivatives Find and simplify the derivative of the following functions.
f(x) = 3x⁴(2x²−1)
Guida verificata passo dopo passo1
Identify the function as a product of two functions: f(x) = u(x) * v(x), where u(x) = 3x⁴ and v(x) = (2x²−1).
Apply the product rule for derivatives, which states that the derivative of a product u(x)v(x) is u'(x)v(x) + u(x)v'(x).
Find the derivative of u(x) = 3x⁴. The derivative, u'(x), is obtained by applying the power rule: u'(x) = 12x³.
Find the derivative of v(x) = 2x²−1. The derivative, v'(x), is obtained by applying the power rule: v'(x) = 4x.
Substitute u(x), u'(x), v(x), and v'(x) into the product rule formula: f'(x) = 12x³(2x²−1) + 3x⁴(4x). Simplify the expression to find the derivative of f(x).
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Derivatives
A derivative represents the rate at which a function changes at a given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific point.
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. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are multiplied together, as in the case of f(x) = 3x⁴(2x²−1).
Video consigliato:
05:18The Product Rule
Simplification of Derivatives
After finding the derivative of a function, simplification is often necessary to express the result in a more manageable form. This may involve combining like terms, factoring, or reducing fractions. Simplifying the derivative helps in understanding the behavior of the function and makes it easier to analyze critical points and concavity.
Video consigliato:
Percorso guidato
05:44Derivatives
Pratica correlata
Domanda del libro di testo
How is lim x🠂0 sin x/x used in this section?
441
views
Domanda del libro di testo
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 3x) / x
365
views
Domanda del libro di testo
Find the derivative of the following functions.
y = x² In x
163
views
Domanda del libro di testo
Use limits to find f' (x) if f(x) = 7x.
233
views
Domanda del libro di testo
Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Find f′(7).
156
views