Skip to main content
Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.5.87.c
Capitolo 3, Problema 3.5.87.c

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.
c. d/dx (sin⁴ x)

Guida verificata passo dopo passo
1
First, recognize that the function sin⁴(x) can be rewritten as (sin(x))⁴. This helps in applying the chain rule effectively.
Apply the chain rule: If you have a function u(x) raised to a power n, the derivative is n * u(x)^(n-1) * u'(x). Here, u(x) = sin(x) and n = 4.
Calculate the derivative of u(x) = sin(x), which is u'(x) = cos(x).
Substitute u(x) = sin(x) and u'(x) = cos(x) into the chain rule formula: 4 * (sin(x))³ * cos(x).
Simplify the expression to get the derivative: 4 * sin³(x) * cos(x). This is the derivative of sin⁴(x) using the chain rule.

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:
4m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Product Rule

The Product Rule is a fundamental differentiation technique used when finding the derivative of a 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 d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x). This rule is essential for calculating derivatives where functions are multiplied together, such as in the case of sin^n x.
Video consigliato:
05:18
The Product Rule

Chain Rule

The Chain Rule is another critical differentiation rule used when dealing with composite functions. It states that if a function y is composed of another function u, such that y = f(u) and u = g(x), then the derivative is given by dy/dx = dy/du * du/dx. In the context of sin^n x, the Chain Rule is necessary to differentiate the inner function (sin x) raised to a power.
Video consigliato:
05:02
Intro to the Chain Rule

Higher Order Derivatives

Higher order derivatives refer to the derivatives of a function taken multiple times. For example, the second derivative is the derivative of the first derivative. In the context of sin^n x, understanding higher order derivatives can be important for analyzing the behavior of the function, such as concavity and points of inflection, especially when applying the Product Rule and Chain Rule in more complex scenarios.
Video consigliato:
02:42
Higher Order Derivatives
Pratica correlata
Domanda del libro di testo

{Use of Tech} Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W=1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh=3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t+4t²−t³ / 9kWh where t=0 corresponds to midnight.

c. Graph the power function and interpret the graph. What are the units of power in this case?

215
views
Domanda del libro di testo

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>

c. (f^-1)'(1)

167
views
Domanda del libro di testo

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x)=(sec^−1 x)/x on [1,∞)

219
views
Domanda del libro di testo

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

c. What is the height of the stone at the highest point?

234
views
Domanda del libro di testo

Computing the derivative of f(x) = e^-x

c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

255
views
Domanda del libro di testo

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t²+19.6t+24.5.

c. What is the height of the stone at the highest point?

183
views