Problem 31 - Thomas Calculus

Question

Find the first and second derivatives of the functions in Exercises \(y=\frac{x^{3}+7}{x}\)

Step-by-Step Solution

Answer

The first derivative is \( y' = 2x - 7x^{-2} \) and the second derivative is \( y'' = 2 + 14x^{-3} \).

Step 1: Rewrite the function

Start by rewriting the function to make it easier to differentiate. The function is given as \( y = \frac{x^3 + 7}{x} \). Separate the terms inside the fraction:\[y = \frac{x^3}{x} + \frac{7}{x} = x^2 + 7x^{-1}\]This will simplify our differentiation process.

Step 2: Find the first derivative

Differentiate each term of the rewritten function with respect to \(x\). Using the power rule, \( \frac{d}{dx} (x^n) = nx^{n-1} \):1. The derivative of \( x^2 \) is \( 2x^{1} = 2x \).2. The derivative of \( 7x^{-1} \) is \( 7(-1)x^{-2} = -7x^{-2} \).Combine these results to find the first derivative:\[y' = 2x - 7x^{-2}\]

Step 3: Find the second derivative

Differentiate the first derivative \( y' = 2x - 7x^{-2} \) once more to find the second derivative:1. The derivative of \( 2x \) is \( 2 \).2. The derivative of \( -7x^{-2} \) is \( -7(-2)x^{-3} = 14x^{-3} \).Combine these results to find the second derivative:\[y'' = 2 + 14x^{-3}\]

Key Concepts

First Derivative

Calculus often involves finding derivatives, which tell us how a function changes at any given point. The first derivative of a function, often denoted as \( y' \) or \( \frac{dy}{dx} \), represents the slope of the tangent line to the function at a particular point. In simpler terms, it tells us the rate of change or the speed of something with respect to another variable.

For example, in the function \( y = \frac{x^3 + 7}{x} \), we started by rewriting the equation as \( y = x^2 + 7x^{-1} \) to make differentiation easier. By breaking it down into simpler terms, the power of each term becomes clear to apply the power rule.

• The derivative of \( x^2 \) is \( 2x \), which indicates that for every unit increase in \( x \), \( y \) increases by \( 2x \).
• For the term \( 7x^{-1} \), its derivative \( -7x^{-2} \) shows that as \( x \) increases, this part of the function decreases, affecting the overall rate of change.

Together, these derivatives combine to form the first derivative \( y' = 2x - 7x^{-2} \), showing us precisely how the function changes at any given value of \( x \). Understanding the first derivative is essential for analyzing functions, predicting trends, and conducting optimizations.

Second Derivative

Once we obtain the first derivative, we can explore the second derivative, which provides insight into the curvature or concavity of the original function. The second derivative, denoted as \( y'' \) or \( \frac{d^2y}{dx^2} \), describes how the rate of change itself changes. Think of it as the "acceleration" of the function.

From our exercise, the first derivative \( y' = 2x - 7x^{-2} \) was differentiated again to get the second derivative. Here's how:

• The second derivative of \( 2x \) reduces to \( 2 \), indicating a constant slope.
• For \( -7x^{-2} \), its derivative is \( 14x^{-3} \), implying that this term affects the change in curvature based on \( x \).

Putting them together, we determine that \( y'' = 2 + 14x^{-3} \). This provides an understanding of how the function behaves beyond the immediate slope, revealing its concave or convex nature. If \( y'' \) is positive at a point, \( y \) is concave up (think a "cup"), while a negative \( y'' \) implies a concave down form (shaped like an "umbrella"). This is pivotal in understanding points of inflection and behavior of the graph over time.

Power Rule

The power rule is a fundamental principle in calculus which simplifies the differentiation of polynomial functions. It states: if you have a term \( x^n \) in your function, its derivative is \( nx^{n-1} \). This rule makes differentiation straightforward for most standard forms.

In our exercise, where we had \( y = x^2 + 7x^{-1} \), both terms were perfect candidates for the power rule. Here's how it was applied:

• For \( x^2 \), applying the power rule gives us: \( 2 \cdot x^{2-1} = 2x \).
• For \( 7x^{-1} \), it's slightly tricky but still follows the rule: \( 7 \cdot (-1) \cdot x^{-1-1} = -7x^{-2} \).

The power rule streamlines the process of finding derivatives for polynomial terms by reducing repetitive calculations into a simple formula. By applying the power rule, you'll find derivatives with confidence and efficiency. Having this skill is crucial not only for calculus but for applying calculus concepts to physics, economics, and beyond.