Problem 13 - Calculus for Biology and Medicine

Question

Use the product rule to find the derivative with respect to the independent variable. $$ g(s)=\left(2 s^{2}-5 s\right)^{2} $$

Step-by-Step Solution

Answer

The derivative of \( g(s)=(2s^2 - 5s)^2 \) is \( 32s^3 - 120s^2 + 100s \).

Step 1: Recognize the Problem Type

We have a function of the form \( g(s) = (f(s))^2 \) where \( f(s) = 2s^2 - 5s \). This is a composition of functions, which suggests using the chain rule due to the squared term. However, since it's a product of itself, the product rule will also be used.

Step 2: Apply the Chain Rule

First, recognize \( g(s) = (f(s))^2 \), meaning we can temporarily write it as \( g(s) = u^2 \), where \( u = 2s^2 - 5s \). The derivative of \( u^2 \) is \( 2u \cdot \frac{du}{ds} \). We need to calculate \( \frac{du}{ds} \) next.

Step 3: Differentiate the Inner Function

Identify the inner function \( f(s) = 2s^2 - 5s \). Its derivative \( \frac{du}{ds} \) is calculated as follows: \( \frac{d}{ds}(2s^2) = 4s \) and \( \frac{d}{ds}(-5s) = -5 \). Therefore, \( \frac{du}{ds} = 4s - 5 \).

Step 4: Combine Derivatives Using Chain Rule

Substitute back into the formula \( 2u \cdot \frac{du}{ds} \). Substituting \( u = 2s^2 - 5s \) and \( \frac{du}{ds} = 4s - 5 \), we achieve the derivative: \( 2(2s^2 - 5s)(4s - 5) \).

Step 5: Simplify the Expression

Expand the expression: \( 2(2s^2 - 5s)(4s - 5) = 4(2s^2 - 5s)(4s - 5) = (8s^2 - 20s)(4s - 5) \). Expanding further: \( (8s^2)(4s) + (8s^2)(-5) + (-20s)(4s) + (-20s)(-5) \). Calculating each term gives: \( 32s^3 - 40s^2 - 80s^2 + 100s \). Combine like terms to get: \( 32s^3 - 120s^2 + 100s \).

Step 6: Final Review and Verification

Rechecking each differentiation and expansion ensures accuracy. The product and chain rules have been correctly applied. Consequently, the correctly derived and simplified expression is \( 32s^3 - 120s^2 + 100s \).

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to find derivatives of composite functions. When you have a function nested inside another function, like \( g(s) = (f(s))^2 \), the chain rule is essential.

Here's a quick refresher:

• You identify the outer function and the inner function. In this case, the outer function is \( u^2 \), and the inner function is \( u = 2s^2 - 5s \).
• Apply the chain rule by differentiating the outer function with respect to the inner function, and then multiply it by the derivative of the inner function.

By applying the chain rule, you calculate the derivative of \((f(s))^2\) as \( 2u \cdot \frac{du}{ds} \). This helps manage complex derivatives and ensures each part of the composite function is addressed.

Derivative

The derivative is a measure of how a function changes as its input changes. In simple terms, it tells you the slope of the function at any point.

For our function \( g(s) = (2s^2 - 5s)^2 \), finding the derivative involves a few steps:

• First, identify the inner function \( f(s) = 2s^2 - 5s \).
• Calculate its derivative: \( \frac{d}{ds}(2s^2 - 5s) = 4s - 5 \).

This derivative tells us how fast \( f(s) \) is changing with respect to \( s \). Now, taking the derivative of the outer function and combining with the inner derivative as per the chain rule, gives a comprehensive view into the function's rate of change.

Function Composition

Function composition involves combining two functions such that the output of one function becomes the input for another. It's like assembling building blocks to create more complex structures.

In this exercise, \( g(s) = (2s^2 - 5s)^2 \) demonstrates function composition:

• The function \( f(s) = 2s^2 - 5s \) is nested inside another function, squaring itself to form \( g(s) \).
• This nesting calls for delicate handling when finding derivatives, often requiring techniques like the chain rule.

Understanding function composition is crucial because it sets the ground for applying the right differentiation methods, enabling you to deconstruct and analyze complex mathematical expressions step by step.