Problem 18 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(s)=\left(\frac{2 s^{2}}{s+1}\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \( h'(s) = 8 \frac{(2s^2)^3(s^2 + 2s)}{(s+1)^5} \).

Step 1: 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.

Step 2: 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 \( u'(s) \).

Step 3: 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 \( w(s) = s+1 \).

Step 4: Differentiate the Numerator and Denominator

Differentiate the numerator: \( v'(s) = \frac{d}{ds}(2s^2) = 4s \). Differentiate the denominator: \( w'(s) = \frac{d}{ds}(s+1) = 1 \).

Step 5: Apply the Quotient Rule

Substitute into the quotient rule:\[ u'(s) = \frac{4s(s+1) - 2s^2(1)}{(s+1)^2} = \frac{4s^2 + 4s - 2s^2}{(s+1)^2} = \frac{2s^2 + 4s}{(s+1)^2} \].

Step 6: Substitute Back into the Chain Rule Expression

We substitute back into the chain rule derivative:\[ h'(s) = 4 \left( \frac{2s^2}{s+1} \right)^3 \cdot \frac{2s^2 + 4s}{(s+1)^2} \].

Step 7: 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.

Key Concepts

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:

• Differentiate the outer function: \( h'(u) = 4(u(s))^3 \).
• Multiply by the inner derivative: \( h'(s) = 4(u(s))^3 imes u'(s) \).

This tells you how changes in the input \( s \) affect the output \( h(s) \). The trick is correctly identifying \( u(s) \) and its derivative \( u'(s) \).

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:

• 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: \( u'(s) = \frac{4s(s+1) - 2s^2}{(s+1)^2} \).

This rule is incredibly helpful when faced with complex rational functions.

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:

• 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.

This process helps manage complex expressions by breaking them into digestible parts, making differentiation straightforward.