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Magazine Chapter 4: Problem 18
In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(s)=\left(\frac{2 s^{2}}{s+1}\right)^{4} $$
Short Answer
Expert verified
The derivative is \( h'(s) = 8 \frac{(2s^2)^3(s^2 + 2s)}{(s+1)^5} \).
Step by step solution
01
Identify the function type
The function given is a composite function involving a rational function and an exponent. It is of the form \( h(s) = \left( u(s) \right)^4 \), where \( u(s) = \frac{2s^2}{s+1} \). We will use the chain rule to differentiate.
02
Apply the Chain Rule
According to the chain rule, the derivative of \( h(s) = (u(s))^4 \) is \( h'(s) = 4(u(s))^3 \cdot u'(s) \). We need to find both \( u(s) \) and \( u'(s) \).
03
Calculate the Derivative of the Inner Function
\( u(s) = \frac{2s^2}{s+1} \) is a rational function. Use the quotient rule for differentiation: if \( u(s) = \frac{v(s)}{w(s)} \), then \( u'(s) = \frac{v'(s)w(s) - v(s)w'(s)}{(w(s))^2} \). Here, \( v(s) = 2s^2 \) and \( w(s) = s+1 \).
04
Differentiate the Numerator and Denominator
Differentiate the numerator: \( v'(s) = \frac{d}{ds}(2s^2) = 4s \). Differentiate the denominator: \( w'(s) = \frac{d}{ds}(s+1) = 1 \).
05
Apply the Quotient Rule
Substitute into the quotient rule:\[ u'(s) = \frac{4s(s+1) - 2s^2(1)}{(s+1)^2} = \frac{4s^2 + 4s - 2s^2}{(s+1)^2} = \frac{2s^2 + 4s}{(s+1)^2} \].
06
Substitute Back into the Chain Rule Expression
We substitute back into the chain rule derivative:\[ h'(s) = 4 \left( \frac{2s^2}{s+1} \right)^3 \cdot \frac{2s^2 + 4s}{(s+1)^2} \].
07
Simplify the Expression
Simplify the expression for the derivative:\[ h'(s) = 8 \frac{(2s^2)^3(s^2 + 2s)}{(s+1)^5} \]. This uses the fact that raising \( (2s^2/(s+1)) \) to the third power involves the cube of the numerator and denominator.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
In calculus, the chain rule is a method for finding the derivative of composite functions. It's especially useful when dealing with functions nested within each other, like in our exercise where we have a function raised to a power.
When you encounter a function of the form \[ h(s) = (u(s))^n, \]you can apply the chain rule to find its derivative. This means multiplying the derivative of the outer function by the derivative of the inner function. For \( h(s) = (u(s))^4 \), we apply:
When you encounter a function of the form \[ h(s) = (u(s))^n, \]you can apply the chain rule to find its derivative. This means multiplying the derivative of the outer function by the derivative of the inner function. For \( h(s) = (u(s))^4 \), we apply:
- Differentiate the outer function: \( h'(u) = 4(u(s))^3 \).
- Multiply by the inner derivative: \( h'(s) = 4(u(s))^3 imes u'(s) \).
Quotient Rule
The quotient rule is a key calculus concept used when differentiating a ratio of two functions. In our problem, \( u(s) = \frac{2s^2}{s+1} \), which is a fraction and perfectly fits the use of the quotient rule.
When you have a function \( \frac{v(s)}{w(s)} \), the derivative using the quotient rule is:\[ u'(s) = \frac{v'(s)w(s) - v(s)w'(s)}{(w(s))^2}. \]Here's a simple breakdown:
When you have a function \( \frac{v(s)}{w(s)} \), the derivative using the quotient rule is:\[ u'(s) = \frac{v'(s)w(s) - v(s)w'(s)}{(w(s))^2}. \]Here's a simple breakdown:
- Differentiate the numerator \( v(s) \): \( v'(s) = 4s \), since the derivative of \( 2s^2 \) is \( 4s \).
- Differentiate the denominator \( w(s) \): \( w'(s) = 1 \), since the derivative of \( s+1 \) is \( 1 \).
- Apply these to the quotient rule formula to find: \( u'(s) = \frac{4s(s+1) - 2s^2}{(s+1)^2} \).
Composite Function Differentiation
Composite function differentiation involves functions that are composed of multiple operations, such as those involving both power and rational components, like our original function.
A composite function has the inner and outer functions behaving like nested layers. For \[ h(s) = \left(\frac{2s^2}{s+1}\right)^4, \] we see that \( u(s) = \frac{2s^2}{s+1} \) is embedded within an exponent function. This combination is what makes it composite.
Here's how you handle such functions:
A composite function has the inner and outer functions behaving like nested layers. For \[ h(s) = \left(\frac{2s^2}{s+1}\right)^4, \] we see that \( u(s) = \frac{2s^2}{s+1} \) is embedded within an exponent function. This combination is what makes it composite.
Here's how you handle such functions:
- First, recognize the "layers" of your function. Separate the outer function from the inner function.
- Use the chain rule to deal with the outer layer while considering the inner layer's influence.
- For rational inner functions, use the quotient rule to find its derivative.
