Problem 43 - Calculus for Biology and Medicine

Question

In Problems \(40-46\), find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. $$ y=\left(1+\left(3 x^{2}-1\right)^{3}\right)^{2} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 36x(1 + (3x^2 - 1)^3)(3x^2 - 1)^2 \)

Step 1: Identify Outer and Inner Functions

Recognize the structure of the function as a composite of multiple functions. Here, the outermost function is \( (u)^2 \), where \( u = 1 + (3x^2 - 1)^3 \). The inner function within this is \( v = (3x^2 - 1)^3 \), which itself is a composite of an inner function \( w = 3x^2 - 1 \).

Step 2: Apply the Outer Function Derivative

Differentiate the outer function \( y = (u)^2 \) with respect to \( u \), which gives \( \frac{dy}{du} = 2u \). Substitute back for \( u \), leading to \( 2(1 + (3x^2 - 1)^3) \).

Step 3: Differentiate Intermediate Function

Find the derivative of the intermediate function \( u = 1 + (v) \), where \( v = (3x^2 - 1)^3 \). The derivative is \( \frac{du}{dv} = 3(3x^2 - 1)^2 \) after applying the power rule to the expression \( (3x^2 - 1)^3 \).

Step 4: Differentiate the Innermost Function

Finally, differentiate the innermost function \( w = 3x^2 - 1 \). The derivative here is \( \frac{dw}{dx} = 6x \), using the power rule \( \frac{d}{dx}[k x^n] = knx^{n-1} \) for \( k = 3 \) and \( n = 2 \).

Step 5: Combine Derivatives Using Chain Rule

Use the chain rule to combine all derivatives: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} \). Substituting the derivatives found: \( \frac{dy}{dx} = 2(1 + (3x^2 - 1)^3) \times 3(3x^2 - 1)^2 \times 6x \).

Step 6: Simplify the Expression

Perform the multiplication: \( 2 \times 3 \times 6 = 36 \). Thus, the expression becomes \( 36x(1 + (3x^2 - 1)^3)(3x^2 - 1)^2 \).

Key Concepts

Composite Functions

Composite functions are functions formed by combining two or more functions. A simple way to think about composite functions is as functions within functions, akin to nested boxes. You apply them sequentially to an input to get a final output.

In calculus, understanding composite functions is essential because many real-world phenomena are modeled using these structures. In the exercise provided, the function \( y = (1 + (3x^2 - 1)^3)^2 \) is a composite function because it involves multiple layers. The outer function needs to be evaluated at the output of the inner function, making it important to separate and deal with each layer individually.

When dealing with composites, it's handy to identify each layer of the functions. For example, in this exercise, identify \( u \) and \( v \):

• Inner function: \( w = 3x^2 - 1 \)
• Intermediate function: \( v = w^3 = (3x^2 - 1)^3 \)
• Outer function: \( u = 1 + v = 1 + (3x^2 - 1)^3 \)

Recognizing these components helps apply differentiation rules correctly, revealing the beauty and complexity of such combinations.

Derivatives

In calculus, a derivative is how we express the rate of change of a function. It's akin to finding the slope or inclination at any given point on a curve. Understanding derivatives is crucial when dissecting composite functions.

When differentiating composite functions, the chain rule is your best ally. For our example, finding \( \frac{d y}{d x} \) involves differentiating layers from the outermost to the innermost, using simple calculus rules at each step. It starts with the outer function's derivative, moves to intermediate functions, and finally deals with the innermost function. Each derivative relates to how variables change throughout the different levels of the composite structure.

The exercise teaches us to:

• Differentiate the outer function \( u^2 \), resulting in \( 2u \).
• Handle the intermediate function's derivative, \( (3x^2 - 1)^3 \), which translates to \( 3(3x^2 - 1)^2 \).
• For the innermost portion, differentiate \( 3x^2 - 1 \) yielding \( 6x \).

These linked derivatives combine through the chain rule to achieve the overall derivative.

Power Rule

The power rule is a fundamental tool in calculus for determining derivatives. It is a specific formula used to differentiate functions of the form \( x^n \), where \( n \) is any real number. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). This simple yet powerful rule is the building block for more complex differentiation tasks, especially in polynomial expressions.

In the original problem, the power rule is employed extensively:

• While differentiating \( u = 1 + (3x^2 - 1)^3 \), you first identify \( v = (3x^2 - 1)^3 \). Applying the power rule, the derivative becomes \( 3(3x^2 - 1)^2 \).
• For \( w = 3x^2 - 1 \), the derivative turns to \( 6x \), again using the power rule.

When combined through the chain rule, the power rule enables computing derivatives of seemingly complicated composite functions by breaking them into simpler parts. Knowing this rule is crucial for tackling higher-level calculus operations efficiently.