Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 40
Differentiate. $$ y=\frac{\frac{2}{3 x}-1}{\frac{3}{x^{2}}+5} $$
Short Answer
Expert verified
The derivative is \[ y' = \frac{ -\frac{6}{x^4} + 6 x^{-3} - \frac{10}{3} x^{-2} }{ \left( \frac{3}{x^{2}} + 5 \right)^2 } \].
Step by step solution
01
Identify the Quotient Rule
The given function is a quotient of two functions: \( y = \frac{u}{v} \). Here, \( u = \frac{2}{3x} - 1 \) and \( v = \frac{3}{x^{2}} + 5 \). We'll use the quotient rule for differentiation, which states: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \].
02
Differentiate the Numerator \( u \)
First, we need to differentiate \( u = \frac{2}{3x} - 1 \). Rewrite \( u \) as \( u = \frac{2}{3} x^{-1} - 1 \). Applying the power rule, we get: \( u' = \frac{2}{3} \cdot (-1) x^{-2} = -\frac{2}{3} x^{-2} \).
03
Differentiate the Denominator \( v \)
Next, differentiate \( v = \frac{3}{x^{2}} + 5 \). Rewrite \( v \) as: \( v = 3 x^{-2} + 5 \). Applying the power rule, we get: \( v' = 3 \cdot (-2) x^{-3} = -6 x^{-3} \).
04
Apply the Quotient Rule
We now have: \( u' = -\frac{2}{3} x^{-2} \) and \( v' = -6 x^{-3} \). Substitute these derivatives into the quotient rule formula: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]. Substitute the expressions: \[ y' = \frac{\left( -\frac{2}{3} x^{-2} \right) \left( \frac{3}{x^2} + 5 \right) - \left( \frac{\frac{2}{3x} - 1}{x^2} \right) \left( -6 x^{-3} \right)}{\left( \frac{3}{x^{2}} + 5 \right)^2} \].
05
Simplify the Expression
Simplify the numerator step-by-step: \[ -\frac{2}{3} x^{-2} \left( \frac{3}{x^2} + 5 \right) = -\frac{2}{3} \cdot \frac{3}{x^4} - \frac{2}{3} \cdot 5 x^{-2} = -\frac{2}{x^4} - \frac{10}{3} x^{-2} \]. Next term: \[ \left( \frac{2}{3x} - 1 \right) \cdot 6 x^{-3} = \frac{2 \cdot 6 x^{-3}}{3x} - 6 x^{-3} = 4 x^{-4} - 6 x^{-3} \]. So, numerator becomes: \[ -\frac{2}{x^4} - \frac{10}{3} x^{-2} - (4 x^{-4} - 6 x^{-3}) = -\frac{6}{x^4} + 6 x^{-3} - \frac{10}{3} x^{-2} \].
06
Final Expression
Combine all elements to write the final differentiated form: \[ y' = \frac{ -\frac{6}{x^4} + 6 x^{-3} - \frac{10}{3} x^{-2} }{ \left( \frac{3}{x^{2}} + 5 \right)^2 } \].
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
calculus
Calculus is a branch of mathematics that focuses on the rates at which quantities change. It is broadly divided into differential calculus and integral calculus. Differential calculus deals with the concept of a derivative, which represents the rate of change of a quantity. Calculus helps us understand changes and motion, facilitating the solving of problems in various scientific fields.
In this exercise, you will delve into differential calculus, specifically using rules to differentiate a complex fraction.
In this exercise, you will delve into differential calculus, specifically using rules to differentiate a complex fraction.
derivative
A derivative represents the rate of change of a function concerning its variable. It's a measure of how a function's output changes as the input changes. Understanding derivatives is crucial as they form the basis of differential calculus. The notation for the derivative of a function \( f \) with respect to \( x \) is \( f'(x) \) or \( \frac{df}{dx} \).
For the given function, differentiating involves finding the rate at which the numerator and denominator change with respect to \( x \). This requires the application of specific differentiation rules.
For the given function, differentiating involves finding the rate at which the numerator and denominator change with respect to \( x \). This requires the application of specific differentiation rules.
quotient rule
The quotient rule is a method for differentiating a function that is the ratio of two differentiable functions. If you have a function \( y = \frac{u}{v} \), where \( u \) and \( v \) are functions of \( x \), the quotient rule states:
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \].
In this exercise, the function \( y \) is a quotient of two functions \( u \) and \( v \). To find the derivative using the quotient rule, you first differentiate both \( u \) and \( v \) individually, then apply the quotient rule formula.
Make sure to simplify the expressions obtained in each step carefully to avoid errors.
\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \].
In this exercise, the function \( y \) is a quotient of two functions \( u \) and \( v \). To find the derivative using the quotient rule, you first differentiate both \( u \) and \( v \) individually, then apply the quotient rule formula.
Make sure to simplify the expressions obtained in each step carefully to avoid errors.
power rule
The power rule is a basic rule for differentiation that makes it easier to find the derivative of polynomials. It states that if you have a function \( f(x) = x^n \), where \( n \) is any real number, its derivative is given by:
\[ f'(x) = n x^{n-1} \].
In this exercise, you encounter terms like \( x^{-1} \) and \( x^{-2} \) which can be differentiated using the power rule. For example, differentiating \( x^{-1} \) would give \( -x^{-2} \). This rule helps in simplifying the differentiation process, especially when combined with the quotient rule for more complex functions.
\[ f'(x) = n x^{n-1} \].
In this exercise, you encounter terms like \( x^{-1} \) and \( x^{-2} \) which can be differentiated using the power rule. For example, differentiating \( x^{-1} \) would give \( -x^{-2} \). This rule helps in simplifying the differentiation process, especially when combined with the quotient rule for more complex functions.
