Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{3}{x^{3}}-\frac{1}{x^{4}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Rules

To solve the problem, we need to identify the derivatives of the given function, which is written in terms of negative exponents. The function is \[ y = rac{3}{x^3} - rac{1}{x^4}. \] Using the power rule for derivatives, which states that the derivative of \( x^n \) is \( nx^{n-1} \), we will apply it to each term individually. Additionally, remember that a negative exponent \( a/x^n \) can be expressed as \( ax^{-n} \).Rewriting the function with negative exponents, we get,\[ y = 3x^{-3} - x^{-4}. \]

Step 2: Differentiate the First Term

Differentiate the first term, \( 3x^{-3} \), using the power rule. The rule gives us:\[ D_x (3x^{-3}) = 3(-3)x^{-3-1} = -9x^{-4}. \] Thus, the derivative of the first term is \( -9x^{-4} \).

Step 3: Differentiate the Second Term

Next, differentiate the second term, \( x^{-4} \), again using the power rule:\[ D_x (x^{-4}) = (-4)x^{-4-1} = -4x^{-5}. \] Thus, the derivative of the second term is \( -4x^{-5} \).

Step 4: Combine the Derivatives

Now, combine the derivatives of each term to find the derivative of \( y \):\[ D_x y = -9x^{-4} + 4x^{-5}. \] We simply add the two derivatives calculated from the previous steps.

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives. It states that the derivative of \( x^n \), where \( n \) is any real number, is \( nx^{n-1} \). This means we take the exponent, multiply it by the coefficient, and then decrease the exponent by one.

This rule is particularly useful because many functions we encounter can be represented as powers of \( x \). By applying the power rule, we simplify the process of differentiation significantly.

For example, if you have a function \( f(x) = 3x^{-3} \), applying the power rule gives you \( f'(x) = 3(-3)x^{-4} = -9x^{-4} \). It's a straightforward method that's invaluable for tackling complex calculus problems efficiently.

Negative Exponents

Negative exponents might seem tricky at first, but they're straightforward once understood. A term like \( x^{-n} \) represents the reciprocal of a positive exponent: \( \frac{1}{x^n} \).

In calculus, using negative exponents can simplify differentiation, especially when applying the power rule. Transforming fractions into terms with negative exponents helps in directly applying calculus rules without extra steps.

For instance, converting the fraction \( \frac{1}{x^4} \) to \( x^{-4} \) makes it easier to apply the power rule: \( D_x(x^{-4}) = -4x^{-5} \). This conversion simplifies notation and calculations, maintaining clarity during problem-solving.

Derivative of a Function

The derivative of a function represents the rate of change of the function's value with respect to a variable, often \( x \). It is a core concept in calculus essential for understanding how functions behave.

To find a derivative, you need to apply differentiation rules, like the power rule, to each term in the function. The goal is to transform the original function into one that conveys the slope of the tangent line at any point.

In the example given, the function \( y = \frac{3}{x^3} - \frac{1}{x^4} \) can be rewritten using negative exponents, becoming \( y = 3x^{-3} - x^{-4} \). Differentiating each term separately gives \( D_x y = -9x^{-4} + 4x^{-5} \), showcasing how the rate of change of each term contributes to the overall derivative.

Calculus Problem-Solving

Solving calculus problems often involves multiple steps and requires a clear understanding of concepts like differentiation. Carefully identifying which rules to apply and simplifying expressions are key in finding accurate solutions.

When confronted with a function, the first step is determining the form that makes it easiest to differentiate, such as using negative exponents. Next, apply differentiation rules methodically to each term.

Always combine derivatives to get the final solution. In our example, each differentiated term is simply summed up: \( -9x^{-4} + 4x^{-5} \). By following these structured steps and understanding core principles like the power rule, students can effectively tackle calculus exercises.