Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\frac{3}{x^{3}}-\frac{1}{x^{4}} $$

Step-by-Step Solution

Answer

The derivative is \( D_x y = -9x^{-4} + 4x^{-5} \).

Step 1: Rewrite the Expression

First, rewrite the given expression using negative exponents so it's easier to differentiate.Given:\[ y = \frac{3}{x^3} - \frac{1}{x^4} \]can be rewritten as:\[ y = 3x^{-3} - x^{-4} \].

Step 2: Differentiate Each Term Separately

Apply the power rule of differentiation to each term in the expression. The power rule states that if \( y = x^n \), then \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \).- For the first term \(3x^{-3}\): \[ D_x(3x^{-3}) = 3 \times (-3)x^{-3-1} = -9x^{-4} \].- For the second term \(x^{-4}\): \[ D_x(x^{-4}) = (-4)x^{-4-1} = -4x^{-5} \].

Step 3: Combine the Results

Add the derivatives of the individual terms together to find the complete derivative of the function:\[ D_x y = -9x^{-4} - (-4x^{-5}) \]which simplifies to:\[ D_x y = -9x^{-4} + 4x^{-5} \].

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used for differentiation. It's a straightforward method that helps us find the derivative of functions involving power expressions. If you have a function of the form \( y = x^n \), the derivative is given by \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \). This means you multiply by the exponent and then reduce the exponent by one.
In the given exercise, the power rule helps differentiate both terms separately:

• For \( 3x^{-3} \), multiply by the exponent \(-3\) and decrease the power by one, resulting in \(-9x^{-4}\).
• For \( x^{-4} \), the process is similar, giving us \(-4x^{-5}\).

This ability to apply the power rule is essential for efficient calculus problem solving.

Negative Exponents

Understanding negative exponents is crucial in calculus, especially when simplifying expressions before differentiation. A negative exponent signifies the reciprocal of the base raised to the opposite positive power. For example, \( x^{-n} = \frac{1}{x^n} \). This transformation allows us to apply calculus rules more easily.
In the original expression \( y = \frac{3}{x^3} - \frac{1}{x^4} \), rewriting with negative exponents yields \( y = 3x^{-3} - x^{-4} \). This conversion simplifies differentiation using the power rule. By handling negative exponents, you can turn complex fractions into manageable terms for calculus operations.
With practice, converting back and forth between standard and negative exponent forms becomes second nature, enhancing your problem-solving toolkit.

Calculus Problem Solving

Tackling calculus problems systematically makes them more approachable. The problem-solving process typically involves several key steps:

• Rewriting the Expression: Convert fractions to negative exponents if needed. This step adds clarity and simplicity to the differentiation process.
• Differentiating Each Term: Apply the power rule to each term separately, which is why understanding both the rule and concepts like negative exponents is critical.
• Combining Results: After differentiating terms independently, combine them to form the complete derivative.

In the example given, rewriting was followed by applying the power rule to each term and combining them into the final derivative \( D_x y = -9x^{-4} + 4x^{-5} \). This methodical approach, grounded in understanding basic principles, facilitates clear and correct calculus solutions.