Domanda del libro di testo
In Exercises 41–58, find dy/dt.
y = ((3t − 4) / (5t + 2))⁻⁵
241
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.9.23
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sin(5√x)
Guida verificata passo dopo passo1
First, identify the function y = sin(5√x). We need to find the derivative dy/dx.
Recognize that the function involves a composition of functions: the sine function and the square root function. Use the chain rule to differentiate.
The chain rule states that if you have a composite function y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Here, f(u) = sin(u) and g(x) = 5√x.
Differentiate the outer function f(u) = sin(u) with respect to u, which gives f'(u) = cos(u).
Differentiate the inner function g(x) = 5√x with respect to x. Recall that √x can be expressed as x^(1/2), so g(x) = 5x^(1/2). The derivative g'(x) = 5 * (1/2) * x^(-1/2).
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:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to x. This is essential for differentiating y = sin(5√x), where the outer function is sine and the inner function is 5√x.
Video consigliato:
05:02
Intro to the Chain Rule
Derivative of Sine Function
The derivative of the sine function, sin(x), is cos(x). This basic derivative rule is crucial when applying the chain rule to differentiate y = sin(5√x). After applying the chain rule, the derivative of the outer function, sin, is needed, which transforms into cos when differentiating.
Video consigliato:
03:53Derivatives of Sine & Cosine
Derivative of Square Root Function
The derivative of the square root function, √x, is 1/(2√x). This rule is important when differentiating the inner function 5√x in the given problem. By applying this derivative rule, we can find the rate of change of the inner function, which is necessary for the application of the chain rule.
Video consigliato:
01:32Derivatives of Other Trig Functions Example 1
Pratica correlata
Domanda del libro di testo
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
2√y = x – y
219
views
Domanda del libro di testo
In Exercises 41–58, find dy/dt.
y = (1 + cos(2t))⁻⁴
257
views
Domanda del libro di testo
In Exercises 43–50, find by implicit differentiation.
y² = x .
x + 1
261
views
Domanda del libro di testo
a. Graph the function
ƒ(x) = { x, -1 ≤ x < 0
{ tan x, 0 ≤ x ≤ π/4.
b. Is ƒ continuous at x = 0?
c. Is ƒ differentiable at x = 0?
Give reasons for your answers.
181
views
Domanda del libro di testo
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sec(x² − 1)
156
views