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Chapter 2: Problem 33

Find the derivatives of the functions. \(p(x)=3\left(\frac{2}{x^{4}}-\frac{2}{x^{3}}\right)\)

Short Answer

Expert verified
The derivative of the function is \( p'(x) = -24x^{-5} + 18x^{-4} \).

Step by step solution

01

Expand the Function

First, rewrite the function in a more convenient form for differentiation. Given function: \[ p(x) = 3 \left( \frac{2}{x^{4}} - \frac{2}{x^{3}} \right) \] Rewrite the fractions as negative exponents: \[ p(x) = 3 \left( 2x^{-4} - 2x^{-3} \right) \]
02

Distribute the Constant

Distribute the constant 3 inside the parenthesis: \[ p(x) = 3 \times 2x^{-4} - 3 \times 2x^{-3} \] Which simplifies to: \[ p(x) = 6x^{-4} - 6x^{-3} \]
03

Apply the Power Rule

To differentiate the function, apply the power rule. The power rule states that for any function of the form \(ax^n\), the derivative is \(a \times n \times x^{n-1}\). Differentiate each term separately: \[ \frac{d}{dx}[6x^{-4}] = 6 \times -4 \times x^{-5} = -24x^{-5} \] \[ \frac{d}{dx}[-6x^{-3}] = -6 \times -3 \times x^{-4} = 18x^{-4} \]
04

Combine the Results

Combine the differentiated terms to get the final derivative: \[ p'(x) = -24x^{-5} + 18x^{-4} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a foundational concept in calculus, essential for differentiation. It simplifies finding the derivative of functions in the form of \( ax^n \). This rule states that to differentiate \( ax^n \), you multiply by the exponent and then reduce the exponent by one. For example, the derivative of \( ax^n \) is \( a \times n \times x^{n-1} \). In practice, if you have \( 5x^3 \), applying the power rule gives \( 15x^2 \) because 5 times 3 is 15, and the exponent 3 becomes 2 after subtracting one.
Differentiation
Differentiation is the process of finding the derivative of a function. It measures the rate at which a function changes at any given point. In our example, we start with the function \( p(x) = 3 \times \frac{2}{x^4} - \frac{2}{x^3} \). We first rewrite this function by expressing the fractions as negative exponents, making it more manageable. After rewriting, we apply the differentiation rules, like the power rule, to each term to find the derivative. The result is a function that represents the slope of the original function at any point.
Negative Exponents
Negative exponents allow for a smoother application of differentiation rules. When dealing with expressions like \( \frac{1}{x^n} \), we rewrite them as \( x^{-n} \). For example, \( \frac{2}{x^4} \) becomes \( 2x^{-4} \). This step simplifies the process of differentiation immensely. In the solution given, the original function \( p(x) \) included fractions that were converted to negative exponents for easier handling. It鈥檚 much simpler to apply the power rule to \( x^{-4} \) than \( \frac{1}{x^4} \), leading to the final derivative calculation.
Calculus for Life Sciences
Calculus is widely used in life sciences to model and understand various biological processes. From population dynamics to enzyme reactions, differential calculus helps in making predictions and crafting precise models. Differentiation, our core task, enables biologists to calculate rates of change and understand how different factors influence these rates. For instance, how fast a population grows or declines, or how concentration levels of a substance change over time, can be understood using derivatives and the foundational concepts we applied like the power rule and handling of negative exponents.

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Most popular questions from this chapter

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