Problem 29 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=(a x+1)^{3} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Step-by-Step Solution

Answer

The derivative is \( 3a(ax+1)^{2} \).

Step 1: Identify the Function Type

The function given is \(f(x)=(a x+1)^{3}\). This is a power function, which takes the form \((u(x))^n\), where \(u(x) = ax + 1\) and \(n = 3\).

Step 2: Apply the Chain Rule

To differentiate \( (u(x))^n \), we'll use the chain rule, which states: \( \frac{d}{dx}((u(x))^n) = n \cdot (u(x))^{n-1} \cdot \frac{du}{dx} \). For our function, this becomes: \( 3 \cdot (ax+1)^{2} \cdot \frac{d}{dx}(ax+1) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u(x) = ax + 1 \). The derivative \( \frac{d}{dx}(ax+1) \) is simply \( a \) because it is a linear function in terms of \(x\).

Step 4: Put It Together

Substitute \( \frac{d}{dx}(ax+1) = a \) and simplify: \[ \frac{d}{dx}(f(x)) = 3 \cdot (ax+1)^{2} \cdot a = 3a(ax+1)^{2} \].

Step 5: Solution Presentation

The differentiated function is \( 3a(ax+1)^{2} \). This is the result for \( \frac{df}{dx} \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It helps you find the derivative of a function that is composed of two or more functions. Imagine you have a function that is wrapped in another function, such as \( f(g(x)) \). Here, \( f \) is the outer function and \( g \) is the inner function. The chain rule tells you how to take the derivative of such a function.

• The chain rule formula is: \[\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x).\]
• First, you differentiate the outer function \( f(x) \) and keep the inner function \( g(x) \) the same.
• Then, multiply by the derivative of the inner function, \( g(x) \).

In our example, the function \((ax + 1)^{3}\) involves the inner function \( ax+1 \) and an outer power function. The chain rule helps us to systematically differentiate it by addressing each layer separately.

Power Function

A power function is a type of function where a variable is raised to a constant power. In its simplest form, it looks like \( x^n \). These functions are very common in math and understanding their properties is crucial for differentiation.

• Example: Given \((u(x))^n\), where \( u(x) \) is a function of \( x \) and \( n \) is a constant.
• The power rule is used for differentiation, which states: \[\frac{d}{dx}(u(x))^n = n \cdot u(x)^{n-1} \cdot \frac{du}{dx}.\]

With the power function, each term \( x \) is raised to the same power. Differentiating such functions involves bringing the power down and reducing the power by one, then multiplying by the derivative of \( u(x) \). In the example, \((ax + 1)^{3}\) behaves as a power function, and we apply this rule to find its derivative.

Derivative

The derivative is a cornerstone concept in calculus, and it measures how a function changes as its input changes. Essentially, it provides the rate at which one quantity shifts concerning another. Derivatives describe the behavior of functions and are pivotal in various fields like physics, engineering, and economics.

• When differentiating a function \( f(x) \), we are finding \( f'(x) \), which is the derivative.
• The derivative of a linear function \( ax + b \) is simply \( a \).
• For more complex functions, techniques like the chain rule and power rule help find derivatives efficiently.

In the example, differentiating \((ax + 1)^{3}\) employs these rules to derive \( 3a(ax+1)^{2} \). This result shows how \( f(x) \) changes as \( x \) changes, translated to a rate of change of the power function combined with the linear function inside it.