Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The derivative is \(-\frac{2}{x^2} + \frac{2}{x^3}\)."

Step 1: Understand the Problem

You need to find the derivative of the function \( y = \frac{2}{x} - \frac{1}{x^2} \) with respect to \( x \). This involves applying differentiation rules to each term separately.

Step 2: Differentiate \( \frac{2}{x} \)

The term \( \frac{2}{x} \) can be rewritten as \( 2x^{-1} \). The derivative of \( x^{-n} \) is \(-nx^{-(n+1)} \). Hence, the derivative of \( 2x^{-1} \) is \(-2x^{-2} \).

Step 3: Differentiate \( \frac{1}{x^2} \)

The term \( \frac{1}{x^2} \) can be rewritten as \( x^{-2} \). The derivative of \( x^{-2} \) is \(-2x^{-3} \).

Step 4: Combine the Derivatives

Combine the derivatives from Steps 2 and 3. The derivative of \( y \) with respect to \( x \) is: \[ D_{x} y = -2x^{-2} + 2x^{-3} \] Simplifying, this gives: \[ D_{x} y = -\frac{2}{x^2} + \frac{2}{x^3} \]

Step 5: Final Answer

The derivative \( D_{x} y = \frac{d}{dx}(\frac{2}{x} - \frac{1}{x^2}) \) is \[ D_{x} y = -\frac{2}{x^2} + \frac{2}{x^3} \].

Key Concepts

Differentiation Rules

Differentiation is a key concept in calculus that allows us to determine the rate at which a function changes. For the function given in the exercise, we apply the differentiation rules to each component separately. For rational function components, such as those in the given exercise, we often see terms represented as powers of the variable, which will allow us to use specific rules for easier calculations. Different rules apply depending on whether components are polynomials, products, or quotients. Understanding each type and using the corresponding rule makes the process of differentiation more straightforward.

For this particular problem, our main rule is the power rule. Additionally, recognizing when to break apart complex fractions and rewrite them as sums or differences can streamline the solution process. Each term in a function is handled individually and then combined for the final result.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. The function in the exercise, \( y = \frac{2}{x} - \frac{1}{x^2} \), is rational. It's expressed as a difference of two simple rational terms.

In rational functions, alterations such as rewriting expressions can assist in identifying simpler forms for applying differentiation techniques. Rewriting \( \frac{2}{x} \) as \( 2x^{-1} \) and \( \frac{1}{x^2} \) as \( x^{-2} \) transform the division into a more familiar power format. This translation to powers is critical for using differentiation rules effectively.

Grasping the structure of rational functions and their components allows simplification into more direct terms, easing the calculation of derivatives.

Power Rule

The power rule is one of the most essential and widely used rules for differentiation. It states that if \( y = x^n \), then the derivative \( y' = nx^{n-1} \). This is a straightforward method where we "bring down" the exponent as a coefficient, then reduce the exponent by one.

In the problem, the function includes \( 2x^{-1} \) and \( x^{-2} \). Applying the power rule:

• For \( 2x^{-1} \), the derivative is \(-2x^{-2}\) by multiplying by the exponent -1 and reducing it by one.
• For \( x^{-2} \), the derivative is \(-2x^{-3}\) following the same process.

Using the power rule effectively reduces even complex-looking functions into manageable components that can be easily differentiated step-by-step.

Simplifying Expressions

After applying the differentiation rules, it's crucial to simplify expressions to clearly communicate the derivative. This involves combining terms where possible and ensuring the result is as concise as it can be.

In our exercise, after finding the derivatives of the individual components, we combine them:

• The derivative of \( y = \frac{2}{x} - \frac{1}{x^2} \) is \( -2x^{-2} + 2x^{-3} \).
• This expression can be rewritten for clarity as \( -\frac{2}{x^2} + \frac{2}{x^3} \).

Combining like terms and simplifying fractions where applicable creates an easier-to-understand expression. Simplification ensures that the final answer is presented in the clearest form, helping to reduce errors in interpretation and further calculation.