Problem 2 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=(4 x+5)^{3}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Outer Function and Inner Function

The given function is presented in a composite form: \( f(x) = (4x + 5)^3 \). Here, the outer function is \( u^3 \) and the inner function is \( u = 4x + 5 \). This is a typical application of the chain rule in differentiation.

Step 2: Differentiate the Outer Function

Differentiate the outer function \( u^3 \) with respect to \( u \). The derivative is \( \frac{d}{du}(u^3) = 3u^2 \).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \( u = 4x + 5 \) with respect to \( x \). The derivative is \( \frac{d}{dx}(4x + 5) = 4 \).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative of a composite function \( f(x) = (g(h(x))) \) is \( f'(x) = g'(h(x)) \cdot h'(x) \). Here, that means substituting \( u = 4x + 5 \) into the derivative of the outer function and multiplying by the derivative of the inner function. Thus, \( f'(x) = 3(4x + 5)^2 \times 4 \).

Step 5: Simplify the Expression

Finally, simplify the expression to get the derivative of the function. \( f'(x) = 12(4x + 5)^2 \).

Key Concepts

Chain Rule

When faced with differentiating composite functions, the chain rule is your best friend. It provides a systematic way to differentiate functions that are composed of other functions. Simply put, the chain rule is used when you need to find the derivative of a function that has another function nested inside it. This is often represented as \( f(g(x)) \). In our given exercise, the function \( f(x) = (4x + 5)^3 \) is a perfect candidate for applying the chain rule, as it consists of an "inner" function, \( h(x) = 4x + 5 \), raised to a power—the "outer" function, \( g(u) = u^3 \).To use the chain rule,

• First, differentiate the outer function with respect to the inner function (treat the inner function as a single variable).
• Then, multiply this by the derivative of the inner function itself.

This method ensures you capture the rate of change throughout the entire composite function, not just one part of it. In our example, it will give us \( f'(x) = 3(4x + 5)^2 \times 4 \), which simplifies to \( 12(4x + 5)^2 \).

Composite Function

Understanding composite functions is essential for applying the chain rule effectively. A composite function is, by definition, a function that is made up of two or more functions. This means each input travels through multiple "layers" of operations. In mathematical terms, it's written as \( f(g(x)) \), where one function is the innermost operation applied to the input \( x \), and the other is the operation applied to the result of the inner function.For the function \( f(x) = (4x + 5)^3 \), think of 4x + 5 as making a result first, and then the exponent 3 applying to that result. So,

• First, you apply the inner function \( g(x) = 4x + 5 \), which adds 5 to the product of 4 and \( x \).
• Then you apply the outer function, which simply cubes the result of the inner function.

Recognizing that a function is composite helps in organizing your differentiation process, specifically by knowing how to apply the chain rule effectively.

Derivative Calculation

Calculating derivatives forms the backbone of differential calculus, focusing on how functions change and providing valuable insights into their behavior. Using the chain rule for derivative calculation simplifies working with composite functions. By addressing each component of the composite, we ensure thorough differentiation.The steps in deriving \( f(x) = (4x + 5)^3 \) involve:

• First, differentiate the outer function \( u^3 \) as \( 3u^2 \).
• Next, differentiate the inner function \( 4x + 5 \) as 4.
• Combine the results: multiply \( 3(4x + 5)^2 \) by 4 to get \( 12(4x + 5)^2 \).

By simplifying further, you can effectively visualize how changing \( x \) impacts the entire function. This entire structured approach helps not only compute derivatives but also better understand what the output signifies for every change in \( x \). Comprehending derivative calculations allows a deeper grasp of how math models shifts and trends, crucial for advanced mathematics and real-world applications.