Problem 221 - Calculus Volume 1

Question

For each of the following exercises, a. decompose each function in the form \(y=f(u)\) and \(u=g(x),\) and b. find \(\frac{d y}{d x}\) as a function of \(x .\) $$y=\left(3 x^{2}+1\right)^{3}$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 18x(3x^2 + 1)^2 \).

Step 1: Decompose the Function

Identify functions \( f(u) \) and \( u=g(x) \) such that \( y=f(u) \) and \( u=g(x) \). In this problem, let \( u = 3x^2 + 1 \) and \( y = u^3 \). So, we have \( f(u) = u^3 \) and \( g(x) = 3x^2 + 1 \).

Step 2: Apply the Chain Rule

To find \( \frac{dy}{dx} \), use the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). First, find \( \frac{dy}{du} \) and \( \frac{du}{dx} \).

Step 3: Differentiate \( y = u^3 \) with Respect to \( u \)

Differentiate \( f(u) = u^3 \) to find \( \frac{dy}{du} \). The derivative of \( u^3 \) is \( \frac{dy}{du} = 3u^2 \).

Step 4: Differentiate \( u = 3x^2 + 1 \) with Respect to \( x \)

Differentiate \( g(x) = 3x^2 + 1 \) to find \( \frac{du}{dx} \). The derivative of \( 3x^2 + 1 \) is \( \frac{du}{dx} = 6x \).

Step 5: Substitute and Solve for \( \frac{dy}{dx} \)

Substitute \( \frac{dy}{du} = 3u^2 \) and \( \frac{du}{dx} = 6x \) into the chain rule formula \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \):\[ \frac{dy}{dx} = 3u^2 \cdot 6x = 18xu^2 \].

Step 6: Express \( \frac{dy}{dx} \) in Terms of \( x \)

Substitute back \( u = 3x^2 + 1 \) into the expression for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = 18x(3x^2 + 1)^2 \].

Key Concepts

Differentiation

Differentiation is one of the fundamental concepts in calculus. It's a process used to compute a derivative, which represents the rate of change of a function with respect to its variables. When you differentiate a function, you obtain a new function that gives the slope of the original function at any point. This is crucial in exploring and understanding the behavior of curves. In our exercise, the function is given as \( y = (3x^2 + 1)^3 \). Differentiation helps us find how \( y \) changes as \( x \) changes, and in this step, we ultimately seek the mathematical expression known as \( \frac{dy}{dx} \).

• Understand that the derivative \( \frac{dy}{dx} \) provides the slope or steepness of the curve at any point \( x \).
• Through differentiation, you can determine critical points, where the function increases or decreases, and identify inflection points.
• It's essential to follow rules and formulas in calculus such as the power rule, chain rule (as in our current example), and others to differentiate correctly.

Continuous learning and practice will enhance your ability to differentiate complex functions smoothly. The more you explore different functions, shapes, and curves, the better understanding of this concept you will develop.

Function Decomposition

Breaking a function down into simpler parts, or function decomposition, is a powerful technique in calculus, especially when dealing with composite functions. The goal is to express a complex function as the composition of two or more simpler functions. This is key to applying rules like the chain rule, which handles differentiation of composite functions.

In our exercise, the function \( y = (3x^2 + 1)^3 \) can be decomposed into two functions:

• \( u = g(x) = 3x^2 + 1 \)
• \( y = f(u) = u^3 \)

We've expressed \( y \) first as a function of \( u \), and then \( u \) as a function of \( x \). Function decomposition simplifies the process of differentiation by allowing us to focus on differentiating each part individually, and later combining the results, usually via the chain rule. By breaking it down, complex operations can be handled in manageable steps, ensuring that errors are minimized and the correct derivative is found.

Derivatives

Derivatives measure how a function changes as its input changes. It is a central concept in calculus, providing insights into the behavior and characteristics of functions. Taking the derivative involves using specific rules to find a function's rate of change.

In this exercise, after decomposing the function, we need to find the derivative of each part:

• First, differentiate \( f(u) = u^3 \) to get \( \frac{dy}{du} = 3u^2 \).
• Then, differentiate \( g(x) = 3x^2 + 1 \) to obtain \( \frac{du}{dx} = 6x \).

To find the total derivative \( \frac{dy}{dx} \), use the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). By substituting \( u = 3x^2 + 1 \) back, we achieve the final expression \( \frac{dy}{dx} = 18x(3x^2 + 1)^2 \). This result shows how \( y \) changes in relation to \( x \) and reflects the overall slope of the function at any given point, allowing us to analyze and predict the behavior of functions effectively.