Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-6, find dw/dt by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=\sin \left(x y z^{2}\right) ; x=t^{3}, y=t^{2}, z=t $$

Step-by-Step Solution

Answer

\( \frac{dw}{dt} = \cos(t^{10}) \cdot (5t^6 + 2t^5) \).

Step 1: Understanding the Function Composition

We want to find \( \frac{dw}{dt} \). The function \( w = \sin(xyz^2) \) is composed with \( x = t^3 \), \( y = t^2 \), and \( z = t \). This means that \( w \) is indirectly dependent on \( t \) through \( x \), \( y \), and \( z \).

Step 2: Apply the Chain Rule

By the Chain Rule, \( \frac{dw}{dt} = \frac{dw}{dx} \cdot \frac{dx}{dt} + \frac{dw}{dy} \cdot \frac{dy}{dt} + \frac{dw}{dz} \cdot \frac{dz}{dt} \). For this, we need to find each of these derivatives.

Step 3: Find Derivative of Function w with respect to x, y, z

First, we differentiate \( w = \sin(xyz^2) \):* \( \frac{dw}{dx} = \cos(xyz^2) \cdot yz^2 \)* \( \frac{dw}{dy} = \cos(xyz^2) \cdot xz^2 \)* \( \frac{dw}{dz} = \cos(xyz^2) \cdot 2xyz \)

Step 4: Find Derivatives of x, y, z with respect to t

Calculate:* \( \frac{dx}{dt} = 3t^2 \) (since \( x = t^3 \))* \( \frac{dy}{dt} = 2t \) (since \( y = t^2 \))* \( \frac{dz}{dt} = 1 \) (since \( z = t \))

Step 5: Substitute and Simplify

Substitute the derivatives from Steps 3 and 4 into the Chain Rule expression:\[ \frac{dw}{dt} = \cos(t^6 \cdot t^2 \cdot t^2) \cdot t^2 \cdot (t \cdot t^2) \cdot 3t^2 + \cos(t^6 \cdot t^2 \cdot t^2) \cdot t^3 \cdot (t \cdot t^2) \cdot 2t + \cos(t^6 \cdot t^2 \cdot t^2) \cdot t^3 \cdot t^2 \cdot 2t \cdot 1 \]Each term simplifies as:\[ = \cos(t^6 t^4) \cdot (t^4 \cdot 3t^2 + t^5 \cdot 2t + t^5 \cdot 2) \]= \[ \cos(t^{10}) \cdot (3t^6 + 2t^6 + 2t^5) \].

Step 6: Simplify Further

Combine like terms in the expression:\[ 3t^6 + 2t^6 + 2t^5 = 5t^6 + 2t^5 \], so the derivative becomes:\( \frac{dw}{dt} = \cos(t^{10}) \cdot (5t^6 + 2t^5) \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a quantity changes. When you differentiate a function, you are essentially measuring how fast something is changing at any given point. This is crucial for understanding patterns of change in mathematical models. In the context of the original exercise, differentiation is used to determine how the function \( w = \sin(xyz^2) \) changes with respect to time \( t \), through the variables \( x \), \( y \), and \( z \).

• To achieve differentiation, calculus provides a set of rules and techniques.
• A primary rule used in differentiation is the Chain Rule, which helps differentiate composite functions.

By differentiating, we find the mathematical derivative \( \frac{dw}{dt} \), enlightening us about the rate of change of \( w \) as \( t \) evolves.

Composite Functions

Composite functions are made when one function is applied to the result of another function. This means the output from one function becomes the input of another. In the exercise, the function \( w = \sin(xyz^2) \) involves the composition of multiple layers of functions, where \( x = t^3 \), \( y = t^2 \), and \( z = t \). Here, each function \( x \), \( y \), and \( z \) is itself dependent on \( t \), making \( w \) a composite function of \( t \).

• Understanding composite functions is essential as they frequently appear in various mathematical and real-world contexts.
• The complexity of composite functions often requires the use of the Chain Rule for differentiation.

In essence, a composite function combines several simple functions into a more complex expression, requiring careful differentiation to determine the overall rate of change.

Mathematical Derivatives

Mathematical derivatives represent the rate of change of a function with respect to a variable. They play an essential role in calculus, underpinning the study of motion, change, and everything that evolves over time.In the initial problem, after applying the Chain Rule, we find derivatives such as: \( \frac{dx}{dt} = 3t^2 \), \( \frac{dy}{dt} = 2t \), and \( \frac{dz}{dt} = 1 \). These derivatives are foundational in constructing the derivative \( \frac{dw}{dt} \).

• Each derivative provides a piece of the puzzle needed to understand the entire system's dynamics.
• Mathematical derivatives allow us to compute exact rates of change in complex systems intertwined with various dependent variables.

Ultimately, understanding how to derive and manipulate mathematical derivatives equips us with powerful tools to model real-world phenomena accurately.