Problem 82 - Calculus for Biology and Medicine

Question

Find the first and the second derivatives of each function. \(f(x)=x^{3}-\frac{1}{x^{3}}\)

Step-by-Step Solution

Answer

The first derivative is \(f'(x) = 3x^2 + 3x^{-4}\); the second derivative is \(f''(x) = 6x - 12x^{-5}\).

Step 1: Identify the function components

The given function is composed of two parts: the polynomial function \(x^3\) and the reciprocal power function \(-\frac{1}{x^3}\). Our goal is to find the derivatives of these individual parts separately.

Step 2: Find the first derivative of the polynomial part

For the polynomial function \(x^3\), use the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Therefore, the derivative of \(x^3\) is \(3x^2\).

Step 3: Find the first derivative of the reciprocal power part

Rewrite \(-\frac{1}{x^3}\) as \(-x^{-3}\), then apply the power rule. The derivative is \(3x^{-4}\), because the power rule yields \(-(-3)x^{-3-1}\).

Step 4: Combine first derivatives

To find the first derivative of the entire function \(f(x) = x^3 - \frac{1}{x^3}\), combine the results: \(f'(x) = 3x^2 + 3x^{-4}\).

Step 5: Differentiate the first derivative to find the second derivative

Using \(f'(x) = 3x^2 + 3x^{-4}\), find the second derivative by differentiating each term individually. The derivative of \(3x^2\) is \(6x\), and the derivative of \(3x^{-4}\) is \(-12x^{-5}\), using the power rule.

Step 6: Combine second derivatives

Combine the second derivatives to get the final expression: \(f''(x) = 6x - 12x^{-5}\).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus to find the derivative of functions that are powers of a variable. It simplifies the process of differentiation and is a cornerstone for understanding calculus operations. The power rule states:

• If you have a function of the form \(x^n\), where \(n\) is a constant, the derivative is given by \(n x^{n-1}\).

For example, if you are differentiating \(x^3\), by applying the power rule, you multiply the exponent 3 by the expression and then reduce the exponent by one, resulting in \(3x^2\). This rule provides a quick and efficient way to differentiate polynomial terms.
It is essential to remember that the power rule applies to all integer, fractional, and negative exponents. This versatility makes it a powerful tool when working through complex functions comprising various terms.
Using the power rule in our exercise, we became able to handle both polynomial and reciprocal power functions smoothly. Understanding and applying the power rule correctly is crucial for solving differentiation problems efficiently.

Polynomial Function

A polynomial function is an expression composed of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Polynomials are some of the most straightforward, yet extensively applied functions in algebra and calculus.

• In this exercise, we encounter the polynomial function \(x^3\), which is a simple monomial of degree 3.
• The degree of the polynomial gives a direct indication of its "shape" and how it behaves as the variable changes.

To differentiate a polynomial, you apply the power rule to each term. For complex polynomial expressions, you can handle each term individually and apply the power rule separately.
Polynomials are straightforward to work with when finding derivatives because each term's structure aligns perfectly with the power rule's requirements. Remember, the derivatives of polynomials are always one degree less than the original function’s degree. Thus, our example \(x^3\) becomes \(3x^2\) after differentiating, fitting in this concept perfectly.

Reciprocal Power Function

Reciprocal power functions typically involve variables raised to negative exponents, or equivalently, expressed as a fraction, like \(\frac{1}{x^n}\). To differentiate such functions, it's often helpful to rewrite them in exponent form.

• For instance, \(-\frac{1}{x^3}\) can be rewritten as \(-x^{-3}\), which allows us to apply the power rule seamlessly.
• Despite its negative exponent, it's treated like any polynomial for differentiation purposes.

Applying the power rule here involves a critical twist: multiplying by the negative of the exponent reduces the exponent by one. Therefore, for \(-x^{-3}\), its differentiation results in \(3x^{-4}\). This showcases the power rule's adaptability for myriad expressions, including those with negative powers.
The reciprocal nature of these functions often leads to solutions that introduce negative or fractional exponents. Understanding reciprocal power functions and their derivatives is crucial, especially as they frequently appear in integrals, limits, and real-world modeling scenarios.