Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=2 x^{-6}+x^{-1} $$

Step-by-Step Solution

Answer

\( D_{x}y = -12x^{-7} - x^{-2} \)

Step 1: Identify the Function Terms

The function given is \( y = 2x^{-6} + x^{-1} \). It consists of two terms: \( 2x^{-6} \) and \( x^{-1} \). We will differentiate these terms separately with respect to \( x \).

Step 2: Differentiate the First Term

For the first term \( 2x^{-6} \), apply the power rule of differentiation: \( \frac{d}{dx}[x^n] = nx^{n-1} \). Differentiating \( 2x^{-6} \), we get:\[ D_{x}(2x^{-6}) = 2 \times (-6)x^{-6-1} = -12x^{-7} \]

Step 3: Differentiate the Second Term

Using the power rule again, differentiate the second term \( x^{-1} \):\[ D_{x}(x^{-1}) = (-1)x^{-1-1} = -x^{-2} \]

Step 4: Combine the Derivatives

Combine the derivatives of the terms to find the derivative of the entire function:\[ D_{x}y = D_{x}(2x^{-6}) + D_{x}(x^{-1}) = -12x^{-7} - x^{-2} \]

Step 5: Write the Final Result

The derivative of \( y \) with respect to \( x \) is:\[ D_{x}y = -12x^{-7} - x^{-2} \]

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus used to differentiate functions that are polynomials. It is a tool that makes finding the derivative of these functions much more straightforward.
The rule states that if you have a function of the form \( x^n \), where \( n \) is any real number, the derivative of this function is \( nx^{n-1} \). This means you bring the exponent down as a multiplier and then reduce the exponent by one.
Here's how to apply it. Consider a term like \( 2x^{-6} \):

• Start by bringing the exponent, which is \(-6\), down to multiply the coefficient (2).
• Subtract one from the exponent to get \(-7\).
• Write the result as \(-12x^{-7}\), since \(2 \times (-6) = -12\).

In this way, the power rule simplifies the process of taking derivatives of polynomial terms by reducing them into simpler expressions.

Differentiation

Differentiation is a core concept in calculus that involves finding the derivative of a function. A derivative represents the rate at which a function changes at any given point. This concept is crucial for understanding how functions behave and is widely applied in various fields such as physics, engineering, and economics.
To differentiate a polynomial function, such as \( y = 2x^{-6} + x^{-1} \), you apply a set of rules designed to systematically take derivatives of each term separately. Here's how to do it:

• Identify each term in the function separately.
• Use the power rule on each of these terms.
• Add the results together to get the derivative of the whole function.

This step-by-step process makes differentiating complex functions more manageable by breaking them into smaller, easily solvable parts.

Calculus

Calculus is an advanced branch of mathematics that deals with how quantities change. It is primarily divided into two main parts: differentiation and integration.
Differentiation focuses on the concept of the derivative, which is used to understand the rate of change or the slope of a function's curve. Calculus enables us to solve problems related to motion, growth, and change within mathematical models.
When differentiating a function like \( y = 2x^{-6} + x^{-1} \), calculus principles allow us to find expressions that describe how \( y \) changes as \( x \) changes. These principles underpin every aspect of how we understand and apply mathematics at a higher level. As foundational as it is complex, calculus is the key that unlocks the dynamic nature of mathematical functions.