Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
![**Problem Statement:**
Given the function:
\[ y = \frac{5x^3 + 2x^2 - 5}{3x^2 + 2} \]
Determine the derivative, \(\frac{dy}{dx}\).
**Explanation:**
This problem involves finding the derivative of a rational function. The function \( y \) is expressed as a quotient of two polynomials. The numerator is \( 5x^3 + 2x^2 - 5 \), and the denominator is \( 3x^2 + 2 \). To find \(\frac{dy}{dx}\), you would typically use the quotient rule for differentiation, which states:
If \( y = \frac{u(x)}{v(x)} \), then:
\[
\frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
Here, \( u(x) = 5x^3 + 2x^2 - 5 \) and \( v(x) = 3x^2 + 2 \). You would first find the derivatives \( u'(x) \) and \( v'(x) \), and then apply the quotient rule formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc7376ae8-f26d-4175-aa22-8c1016731c6d%2Ff83f9ec6-0d2e-4842-b6fc-6f5e62e37d51%2F575tyvo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the function:
\[ y = \frac{5x^3 + 2x^2 - 5}{3x^2 + 2} \]
Determine the derivative, \(\frac{dy}{dx}\).
**Explanation:**
This problem involves finding the derivative of a rational function. The function \( y \) is expressed as a quotient of two polynomials. The numerator is \( 5x^3 + 2x^2 - 5 \), and the denominator is \( 3x^2 + 2 \). To find \(\frac{dy}{dx}\), you would typically use the quotient rule for differentiation, which states:
If \( y = \frac{u(x)}{v(x)} \), then:
\[
\frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
Here, \( u(x) = 5x^3 + 2x^2 - 5 \) and \( v(x) = 3x^2 + 2 \). You would first find the derivatives \( u'(x) \) and \( v'(x) \), and then apply the quotient rule formula.
Expert Solution
Step 1
Consider the provided equation,
Find the derivative of the equation with respect to .
Apply the quotient rule, .
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