Problem 214 - Calculus Volume 1

Question

Given \(y=f(u)\) and \(u=g(x),\) find \(\frac{d y}{d x}\) by using Leibniz's notation for the chain rule: \(\frac{d y}{d x}=\frac{d y}{d u} \frac{d u}{d x}\). $$ y=3 u-6, u=2 x^{2} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = 12x \)

Step 1: Identify Functions

We have two functions given: \ 1. \( y = 3u - 6 \) \ 2. \( u = 2x^2 \) \ We need to find the derivative \( \frac{dy}{dx} \) using the chain rule.

Step 2: Find \( \frac{d u}{d x} \)

Calculate the derivative of \( u \) with respect to \( x \): \ \( u = 2x^2 \) \ Differentiating \( u \) with respect to \( x \) gives: \ \( \frac{du}{dx} = \frac{d}{dx}(2x^2) = 4x \)

Step 3: Find \( \frac{d y}{d u} \)

Calculate the derivative of \( y \) with respect to \( u \): \ \( y = 3u - 6 \) \ Differentiating \( y \) with respect to \( u \) gives: \ \( \frac{dy}{du} = \frac{d}{du}(3u - 6) = 3 \)

Step 4: Apply the Chain Rule

Apply Leibniz's chain rule formula: \ \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \) \ Substitute from the previous steps: \ \( \frac{dy}{dx} = 3 \cdot 4x = 12x \)

Step 5: Conclude

The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = 12x \). This is our final result.

Key Concepts

Derivative

In calculus, a *derivative* measures how a function changes as its input changes. Imagine you're riding a bike along a path—the derivative is like the speedometer showing your speed at each point. It tells us how quickly the value of a function is changing at any given point.

When you have a function like \( y = 3u - 6 \) or \( u = 2x^2 \), calculating the derivative involves determining how each variable affects the function. If you want to find \( \frac{dy}{dx} \), which is how \( y \) changes when \( x \) changes, you'll often use a kind of math magic—the chain rule—to break the task into simpler pieces.

Leibniz's Notation

*Leibniz's notation* is commonly used to express derivatives, indicating which variable we differentiate with respect to. Named after the mathematician Gottfried Wilhelm Leibniz, this notation clearly marks the variable of interest, such as \( \frac{dy}{dx} \) indicating the derivative of \( y \) concerning \( x \).

In our task, to find \( \frac{dy}{dx} \), we rely on the chain rule in Leibniz's notation: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). This approach highlights the intermediate variable \( u \), showing how changes in \( x \) affect \( u \), and subsequently how changes in \( u \) affect \( y \).

This setup simplifies understanding how each part of the function transforms the input to yield the desired outcome.

Differentiation

*Differentiation* is the process of calculating a derivative. It's an essential tool in calculus that helps us understand and describe the real world. Whether analyzing the movement of celestial bodies or predicting economic trends, differentiation provides insights into how things change. 

Consider the function \( u = 2x^2 \). Through differentiation, \( \frac{du}{dx} = 4x \), we measure how \( u \) changes as \( x \) varies. This process shows that for every slight increase in \( x \), \( u \) increases by \( 4x \), reflecting the steep curve of a parabola.

Similarly, differentiating the function \( y = 3u - 6 \) gives us \( \frac{dy}{du} = 3 \). It tells us that \( y \) increases by 3 units for every unit increase in \( u \). By chaining these together, you apply the difference in \( x \) to find the change in \( y \).

Composite Functions

When dealing with *composite functions*, you're essentially handling functions nested within each other, like a Russian doll. Here, \( y = f(u) \) is a function of \( u \), and \( u = g(x) \) is a function of \( x \). These linked functions transform \( x \) through \( u \) to produce \( y \).

Calculating \( \frac{dy}{dx} \) for composite functions involves decomposing them into easier tasks—finding \( \frac{du}{dx} \) and \( \frac{dy}{du} \).

• First, consider \( g(x) = 2x^2 \), which maps \( x \) to \( u \), yielding \( \frac{du}{dx} = 4x \).
• Next, analyze \( f(u) = 3u - 6 \), mapping \( u \) to \( y \), for which \( \frac{dy}{du} = 3 \).
• Finally, the chain rule stitches these derivatives together: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 12x \).

This sequence allows us to unravel even complex equations methodically, helping to simplify and understand the transformations happening at each stage.