Problem 2 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=(4 x+5)^{3} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = (4x+5)^3 \) is \( 12(4x+5)^2 \).

Step 1: Identify the Differentiation Rule

We need to find the derivative of the function \( f(x) = (4x+5)^3 \) with respect to \( x \). This function is in the form \( (u(x))^n \) where \( u(x) = 4x + 5 \) and \( n = 3 \). For this, we will use the Chain Rule.

Step 2: Apply the Chain Rule

The Chain Rule states that \( \frac{d}{dx}[u(x)^n] = n \cdot u(x)^{n-1} \cdot \frac{du}{dx} \). Here, \( u(x) = 4x + 5 \). So we first find \( \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

Find the derivative of \( u(x) = 4x + 5 \) with respect to \( x \), which is \( \frac{du}{dx} = 4 \).

Step 4: Substitute in the Chain Rule Formula

Substitute \( n = 3 \), \( u(x) = 4x + 5 \), and \( \frac{du}{dx} = 4 \) into the Chain Rule formula:\[ \frac{d}{dx}[(4x + 5)^3] = 3 \cdot (4x + 5)^{3-1} \cdot 4 \]

Step 5: Simplify the Expression

Simplify the expression: \[ \frac{d}{dx}[(4x + 5)^3] = 3 \cdot 4 \cdot (4x + 5)^2 = 12(4x + 5)^2 \]

Key Concepts

Chain Rule

When we want to differentiate a composite function, like in our exercise, the Chain Rule is the tool we use. It helps us find the derivative of functions that are "nested" within each other. For example, if a function is in the form \((u(x))^n\), we apply the Chain Rule.
The Chain Rule can be expressed as follows:

• First, differentiate the outside function while keeping the inner function constant.
• Then, multiply the result by the derivative of the inner function.

This two-step process allows us to effectively "chain" the derivatives together, hence the name "Chain Rule". It simplifies the differentiation of complex expressions, making it manageable. In essence, it breaks down the task of differentiation into simpler parts and combines them neatly.

Derivative of a Function

A derivative measures how a function changes as its input changes. It tells us the rate of change or the slope of a function at any given point. When you hear someone mention "differentiation," they are talking about the process of finding this derivative. In mathematical terms, if you have a function \( f(x) \), the derivative is represented as \( f'(x) \) or \( \frac{df}{dx} \).
To compute the derivative, we apply different rules such as the Power Rule, Product Rule, Quotient Rule, and the Chain Rule, depending on the form of the function. By practicing with various functions, you can become adept at applying these rules and finding the derivatives with ease.

Polynomial Differentiation

Differentiating polynomial functions is often a straightforward process because it mainly involves applying the Power Rule. Polynomials are functions like \( ax^n + bx^{n-1} + \, \ldots \, + c \), consisting of multiple terms with exponents.
The Power Rule, one of the simplest differentiation rules, states: "To differentiate the term \( x^n \), multiply by the exponent and reduce the exponent by one." So, \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \). For any polynomial, you apply this rule term by term.
Once you are familiar with the Power Rule, polynomial differentiation becomes a matter of practicing until all equivalent expressions come naturally. Differentiating expressions like \((4x + 5)^3\) might look like they involve polynomials, but because they are composites, they do require the Chain Rule as well.