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 is \(g'(s) = 4(2s^2 - 5s)(4s - 5)\).

Step 1: Identify the Structure

The function given is \(g(s) = (2s^2 - 5s)^2\). The function inside the square, \(2s^2 - 5s\), needs to be identified as a product of two expressions.

Step 2: Apply the Chain Rule

To differentiate \(g(s) = (2s^2 - 5s)^2\), apply the chain rule. First, set \(u = 2s^2 - 5s\), so that \(g(s) = u^2\). According to the chain rule, the derivative \(g'(s) = 2u \, \cdot \, \frac{du}{ds}\).

Step 3: Differentiate the Inner Function

Find the derivative of the inner function \(u = 2s^2 - 5s\). The derivative is \(\frac{du}{ds} = 4s - 5\).

Step 4: Combine Results Using Chain Rule

Substitute back for \(u = 2s^2 - 5s\) and \(\frac{du}{ds} = 4s - 5\) in the expression from Step 2: \(g'(s) = 2(2s^2 - 5s)(4s - 5)\).

Step 5: Simplify the Expression

Now simplify \(g'(s) = 2(2s^2 - 5s)(4s - 5)\) to get the final derivative. Use distribution: \(g'(s) = 2\left((8s^3 - 10s^2) - (20s^2 - 25s)\right)\), which simplifies to \(g'(s) = 4(2s^2 - 5s)(4s - 5)\).

Key Concepts

Chain Rule

When faced with a function inside another function, such as \(g(s) = (2s^2 - 5s)^2\), we can use the chain rule to find its derivative.
The chain rule is a powerful tool for differentiating composite functions. A composite function is simply one that is made up of two or more functions.
This rule helps when a function is nested inside another, allowing you to efficiently compute the derivative by considering the inner and the outer functions separately.

• Identify the outer function and the inner function.
• Differentiating the outer function first, then the inner function.
• Multiply both derivatives for the final result.

For our function, the outer function is \(u^2\) and the inner function is \(u = 2s^2 - 5s\).
Apply the chain rule, differentiating the outer function \(u^2\) gives \(2u\), and the derivative of the inner function \(u\) results in \(4s - 5\). Multiply these two derivatives together: \(g'(s) = 2u \cdot \frac{du}{ds}\).

Derivative

A derivative tells us how a function changes as its input changes. It's essentially the rate of change or slope at any point on the function.
For any given function, the derivative is a new function that describes the behavior of the original function.In the exercise context, you find the derivative \(g'(s)\) by differentiating \(g(s) = (2s^2 - 5s)^2\) using derivative rules.
Since our function is a composite function, the chain rule is used to first find the derivative of the inside function \(u = 2s^2 - 5s\).
The derivative, \(\frac{du}{ds} = 4s - 5\), tells us the rate at which \(u\) changes with respect to \(s\).

• Determine how one quantity changes as another changes.
• Use formulas to establish relationships between variables.

Using the chain rule, the derivative of the overall function is \(g'(s) = 2(2s^2 - 5s)(4s - 5)\), indicating the rate of change of \(g(s)\) with respect to \(s\).

Simplification

Simplification is an important step after finding derivatives to make expressions easier to handle and understand.
We simplify mathematical expressions to get them into their most concise form, making calculations easier.
In this exercise, upon obtaining the derivative using the chain rule, we get the expression \(g'(s) = 2(2s^2 - 5s)(4s - 5)\).

• Distribute terms for simplification.
• Group and combine like terms to reduce complexity.

By distributing and combining terms, the expression becomes more compact and manageable. The process involves multiplying terms together:- Multiply \(2 \cdot (8s^3 - 10s^2 - 20s^2 + 25s)\).- Simplify further to \(4(2s^2 - 5s)(4s - 5)\) for a clear result.
After simplification, the derivative is easier to interpret, providing relevant insights into the function behavior.