Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{3}{x^{3}}+x^{-4} $$

Step-by-Step Solution

Answer

\( D_x y = -9x^{-4} - 4x^{-5} \)

Step 1: Identify the Function Components

The given function is \( y = \frac{3}{x^3} + x^{-4} \). This can be written in terms of power functions as \( y = 3x^{-3} + x^{-4} \).

Step 2: Differentiate the First Term

Use the power rule \( D_{x} x^{n} = nx^{n-1} \) for differentiation. For \( 3x^{-3} \), the derivative is \( D_x(3x^{-3}) = -9x^{-4} \).

Step 3: Differentiate the Second Term

Similarly, use the power rule on \( x^{-4} \). Thus, \( D_x(x^{-4}) = -4x^{-5} \).

Step 4: Combine the Derivatives

After finding the derivative of each term, combine them to get \( D_x y = -9x^{-4} - 4x^{-5} \).

Key Concepts

Power Rule

In calculus, the power rule is a fundamental tool used to find the derivative of functions that are power functions. A power function is any function of the form \( x^n \), where \( n \) is a real number. The power rule is expressed as follows: for any power function \( x^n \), the derivative is given by \( nx^{n-1} \). This rule is essential because it allows us to quickly and easily differentiate functions with exponents.
For instance, consider the function \( 3x^{-3} \). Applying the power rule, we take the exponent \(-3\), multiply it by the coefficient \(3\), and then decrease the exponent by one, resulting in \( -9x^{-4} \). This illustrates how straightforward the application of the power rule can make the process of differentiation, especially for functions that can be expressed as sums of power terms.
Knowing when and how to apply the power rule can significantly simplify the differentiation process, helping solve exercises like the one given.

Derivative

The derivative of a function describes how the function's value changes as the input value changes. It is a core concept in calculus, providing insight into rates of change and the slope of a function's graph at any point. For a given function \( y \), the derivative \( D_x y \) represents the rate of change of \( y \) with respect to \( x \).
To find the derivative of a complex function, we often break it down into simpler components. For instance, in the given function \( y = \frac{3}{x^3} + x^{-4} \), transforming it into \( y = 3x^{-3} + x^{-4} \) allows for the power rule to be applied to each term independently. After differentiating each term, we combine the results to obtain the overall derivative: \( D_x y = -9x^{-4} - 4x^{-5} \).
This gives us a new function that precisely predicts the instantaneous rate at which the original function \( y \) changes, crucial for applications across physics, engineering, and beyond.

Functions

Functions are mathematical expressions that relate a set of inputs to a set of outputs. In calculus, understanding how functions behave under various operations, such as differentiation, is critical. The given exercise focuses on composite functions—specifically, functions made up of terms like \( \frac{3}{x^3} \) and \( x^{-4} \), which are both instances of power functions.
By recognizing that the function can be rewritten in terms of power functions \( 3x^{-3} + x^{-4} \), we simplify the process of applying calculus rules, such as the power rule. Functions like these highlight how operations such as differentiation depend heavily on identifying the forms and structures within the function.
As such, functions serve as the foundation for much of algebra and calculus, helping us to model and solve real-world problems by capturing the relationship between varying quantities in a consistent manner.