Problem 9 - Calculus for Biology and Medicine

Question

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=(3 x-1)^{2} $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = (3x - 1)^2\) is \(f'(x) = 18x - 6\).

Step 1: Recognize the Function

Identify that the function given is a composite function of the form \((g(x))^n\) where \(g(x) = 3x - 1\) and \(n = 2\). This indicates that the power rule and chain rule can be applied to find the derivative.

Step 2: Apply the Chain Rule

Use the chain rule on \(f(x) = (3x - 1)^2\). According to the chain rule, the derivative of \((g(x))^n\) is \(n(g(x))^{n-1} \cdot g'(x)\). Identify \(n = 2\) and \(g(x) = 3x - 1\).

Step 3: Differentiate the Inner Function

Find \(g'(x)\) where \(g(x) = 3x - 1\). The derivative of \(g(x)\) is \(g'(x) = 3\) because the derivative of \(3x\) is 3 and the derivative of a constant \(-1\) is 0.

Step 4: Apply the Chain Rule Formula

Substitute back into the chain rule formula: \(f'(x) = 2(3x - 1)^{2-1} \cdot 3\). Simplify this to get \(f'(x) = 2(3x - 1) \cdot 3\).

Step 5: Simplify the Expression

Complete the simplification: \(f'(x) = 2 \cdot 3 \cdot (3x - 1) = 6(3x - 1)\). This expands to \(18x - 6\).

Step 6: Write the Final Solution

The derivative of \(f(x) = (3x - 1)^2\) is \(f'(x) = 18x - 6\).

Key Concepts

Chain Rule

The chain rule is a powerful technique used in calculus for finding the derivative of a composite function. Composite functions are functions of functions, like when one function is nested inside another. In this exercise, the function is given as \( f(x) = (3x - 1)^2 \), where the inner function \( g(x) = 3x - 1 \) and the outer function raises the result to a power of 2. To apply the chain rule, follow these steps:

• Differently, identify the inside function ( \( g(x) = 3x - 1 \)).
• Calculate its derivative, \( g'(x) = 3 \).
• Recognize the outer function as \( u^2 \), with \( u = g(x) \).
• Find the derivative of the outer function: \( 2u \).

Now, apply the chain rule, which is: \[\frac{du}{dx} = 2(g(x))^{2-1} \cdot g'(x) = 2(3x - 1) \cdot 3.\]
This method systematically breaks down complex derivatives into simpler parts.

Product Rule

Though not directly required here, understanding the product rule is crucial when dealing with derivatives involving multiplication of functions. The product rule states:

• If you have two functions multiplied together \( u(x) \) and \( v(x) \), then the derivative is given by:\[\frac{d}{dx}(u \cdot v) = u'v + uv' \].
• This involves finding the derivative of the first function and multiplying it by the second, then adding the product of the second function's derivative multiplied by the first function.

Had our original function been a product rather than a power, like \( (2x)(3x-1)^2 \), the product rule would have been essential. Combining with the chain rule would ensure accurate differentiation by breaking the problem into manageable pieces.
While the chain rule simplifies composites, the product rule ensures that products of functions are dealt with correctly.

Derivative Simplification

After applying the chain rule (and potentially the product rule), simplifying the resulting expression is crucial for clarity and utility. In our exercise:

• The initial application of the chain rule yields: \( f'(x) = 2(3x - 1) \cdot 3 \).
• Recognize each factor's role and simplify: multiply the constants together, \( 2 \times 3 \), and leave \( (3x - 1) \) as it is.
• This gives us \( f'(x) = 6(3x - 1) \), which can be further simplified by distributing the 6.

Performing this distribution results in: \[f'(x) = 18x - 6\].
Simplification doesn't just make the expression more readable; it also makes evaluating the derivative at specific points much easier. By breaking down and reassembling the problem effectively, simplification follows naturally as an important final step in differentiation.