Problem 10 - 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 \( f'(x) = 3x^2 - 3x^{-4} \), and the second derivative is \( f''(x) = 6x + 12x^{-5} \).

Step 1: Identify the Function Components

The function provided is \( f(x) = x^3 + \frac{1}{x^3} \). Here, we have two components: the term \( x^3 \) and the term \( \frac{1}{x^3} \). We will calculate the derivative of each component separately.

Step 2: Differentiate the First Term

The first term of the function is \( x^3 \). The derivative of \( x^n \) is \( nx^{n-1} \). Applying this to \( x^3 \), the derivative is \( 3x^{3-1} = 3x^2 \).

Step 3: Differentiate the Second Term

The second term is \( \frac{1}{x^3} \), which can be rewritten as \( x^{-3} \). Using the power rule again, the derivative of \( x^{-3} \) is \(-3x^{-3-1} = -3x^{-4} \).

Step 4: Combine First Derivatives

Combine the derivatives of both terms to get the first derivative of the function:\[ f'(x) = 3x^2 - 3x^{-4} \].

Step 5: Differentiate the First Derivative

Now find the second derivative. Start by differentiating \( 3x^2 \), which gives \( 6x \).

Step 6: Differentiate the Second Derivative Term

Differentiate \( -3x^{-4} \). This gives \( 12x^{-5} \) by applying the power rule \( d/dx(-3x^{-4}) = -3(-4)x^{-4-1} = 12x^{-5} \).

Step 7: Combine Second Derivatives

Combine the second derivatives of each term to arrive at the complete second derivative:\[ f''(x) = 6x + 12x^{-5} \].

Key Concepts

Derivatives

Derivatives are a fundamental tool in calculus, helping us understand how a function changes at any given point. The process involves finding the rate at which the function's output value changes as we make small alterations to the input variable. In mathematical terms, the derivative of a function at a given point is the slope of the tangent line to the function's graph at that point. This concept is essential for analyzing the behavior and trends of functions.

Derivatives can be utilized in a wide array of applications, from calculating velocities in physics to finding the optimal solutions in economics. It's like asking, "How fast or slow is something changing?"

In our example, the function is composed of two terms:

• A polynomial term: \( x^3 \)
• A rational term: \( \frac{1}{x^3} \), rewritten as \( x^{-3} \) using negative exponents for easier derivative calculation.

By finding derivatives, we can examine how these individual terms contribute to the rate of change of the entire function.

Power Rule

The power rule is a straightforward method for taking derivatives of polynomial functions or terms that can be expressed as powers of x. It simplifies the differentiation process significantly. The general rule states that if you have a function of the form \( x^n \), its derivative will be \( nx^{n-1} \).

This rule is incredibly useful for both positive and negative exponents:

• For the term \( x^3 \), applying the power rule results in \( 3x^{3-1} = 3x^2 \).
• The term \( \frac{1}{x^3} \) is rewritten as \( x^{-3} \). Applying the power rule gives us \(-3x^{-3-1} = -3x^{-4} \).

The power rule speeds up the differentiation process and is a foundational concept for handling more complex functions. By mastering this rule, you can tackle diverse calculus problems with confidence.

Second Derivative

The second derivative provides insight into the curvature or concavity of the function's graph. While the first derivative helps us determine the rate of change, the second derivative tells us more about the nature of that change. It's like asking, "Is the change itself changing?"

In practical terms:

• If the second derivative is positive for a particular interval, the function is concave up, indicating that the function is increasing at an increasing rate.
• If negative, the function is concave down, showing it is increasing at a decreasing rate.

To find the second derivative in our example:

• Differentiate the first term of the first derivative, \( 3x^2 \), to get \( 6x \).
• Differentiate the second term, \(-3x^{-4} \), to obtain \( 12x^{-5} \).
• Combine these results to achieve the second derivative: \( f''(x) = 6x + 12x^{-5} \).

By understanding second derivatives, you can fully grasp the intricacies of a function's behavior, paving the way to solving complex calculus problems more effectively.