Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 27: Problem 17
Find the derivatives of the given functions. $$y=\frac{2 e^{3 x}}{4 x+3}$$
Short Answer
Expert verified
The derivative is \( y' = \frac{24xe^{3x} + 10e^{3x}}{(4x + 3)^2} \).
Step by step solution
01
Identify the Components of the Function
The given function is \( y = \frac{2e^{3x}}{4x + 3} \). This is in the form of a quotient where the numerator \( u = 2e^{3x} \) and the denominator \( v = 4x + 3 \). The quotient rule for derivatives will be used here.
02
Apply the Quotient Rule
The quotient rule states that if you have a function \( \frac{u}{v} \), the derivative \( y' \) is given by \( \frac{v \cdot u' - u \cdot v'}{v^2} \). We need to find \( u' \) and \( v' \).
03
Differentiate the Numerator
Find \( u' \). Since \( u = 2e^{3x} \), use the chain rule. The derivative of \( e^{3x} \) is \( 3e^{3x} \), so \( u' = 2 \cdot 3e^{3x} = 6e^{3x} \).
04
Differentiate the Denominator
Find \( v' \). The function \( v = 4x + 3 \) is linear, so its derivative is the coefficient of \( x \), which is 4. Thus, \( v' = 4 \).
05
Substitute into the Quotient Rule
Substitute \( u' \), \( v \), \( u \), and \( v' \) into the quotient rule formula: \[ y' = \frac{(4x + 3)(6e^{3x}) - (2e^{3x})(4)}{(4x + 3)^2} \]
06
Simplify the Expression
First, expand the expression in the numerator: \[ (4x + 3)(6e^{3x}) = 24xe^{3x} + 18e^{3x} \]Then:\[ - (2e^{3x})(4) = -8e^{3x} \]Combine these terms in the numerator:\[ 24xe^{3x} + 18e^{3x} - 8e^{3x} = 24xe^{3x} + 10e^{3x} \]Thus, \[ y' = \frac{24xe^{3x} + 10e^{3x}}{(4x + 3)^2} \].
07
Final Derivative
The derivative of the function is: \[ y' = \frac{24xe^{3x} + 10e^{3x}}{(4x + 3)^2} \]. There are no further simplifications possible.
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.
Quotient Rule
The quotient rule is a technique in calculus used to find the derivative of a function that is the ratio of two differentiable functions. It's a foundational tool when dealing with functions expressed as fractions. For a function \(\frac{u}{v}\), where both \(u\) and \(v\) are differentiable, the derivative is given by the formula:
- \(y' = \frac{v \cdot u' - u \cdot v'}{v^2}\)
Chain Rule
The chain rule is essential when dealing with composite functions, where one function is nested inside another. To differentiate such functions, the chain rule provides a systematic method. Consider a composite function \(f(g(x))\), its derivative is:
- \(f'(g(x)) \cdot g'(x)\)
Differentiation
Differentiation is the mathematical process used to find the rate of change of a function. In essence, it's about understanding how a function behaves when its input changes. Applying differentiation involves finding the derivative, which can tell you how steep a curve is at any given point. This is useful when wanting to understand the behavior of a function like \(\frac{2e^{3x}}{4x+3}\).
- Identify the type of function you are dealing with.
- Apply appropriate rules like the chain rule or quotient rule.
Exponential Function
Exponential functions are a class of functions where the variable is in the exponent. They have specific characteristics, most notably continuous growth or decay. When differentiating an exponential function of the form \(e^{kx}\), where \(k\) is a constant, the differentiation process utilizes the chain rule.
- The derivative of \(e^{kx}\) is \(ke^{kx}\).
