Problem 45 - University Calculus: Early Transcendentals

Question

Find the first and second derivatives of the functions. $$y=\frac{x^{3}+7}{x}$$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Function in a Simpler Form

The given function is \( y = \frac{x^3 + 7}{x} \). Begin by simplifying this function to make differentiation straightforward. Divide each term in the numerator by the denominator: \( y = x^2 + \frac{7}{x} = x^2 + 7x^{-1} \).

Step 2: Find the First Derivative

Now differentiate the function \( y = x^2 + 7x^{-1} \) with respect to \( x \). Use the power rule: \( \frac{d}{dx}[x^n] = nx^{n-1} \). For the first term, \( \frac{d}{dx}[x^2] = 2x \). For the second term, \( \frac{d}{dx}[7x^{-1}] = -7x^{-2} \). Thus, the first derivative is \( \frac{dy}{dx} = 2x - \frac{7}{x^2} \).

Step 3: Find the Second Derivative

Differentiate the first derivative \( \frac{dy}{dx} = 2x - 7x^{-2} \). Again use the power rule. For the first term, \( \frac{d}{dx}[2x] = 2 \). For the second term, \( \frac{d}{dx}[-7x^{-2}] = 14x^{-3} \). So the second derivative is the combination of these: \( \frac{d^2y}{dx^2} = 2 + \frac{14}{x^3} \).

Key Concepts

Understanding The Power Rule

The power rule is one of the simplest and most powerful techniques in calculus for finding derivatives. It's used to differentiate functions of the form \(x^n\), where \(n\) is any real number. To apply the power rule, you reduce the power by one and multiply by the original exponent:

• If you have a function \(x^n\), the derivative is \(nx^{n-1}\).

For example, in the function \(y = x^2 + 7x^{-1}\), the term \(x^2\) applies the power rule to become \(2x\). Similarly, \(7x^{-1}\) becomes \(-7x^{-2}\) because we decrease the exponent by one and multiply by 7.
It's a fundamental tool for finding how quantities change and is applied in various scientific fields.

Basics of Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change or the slope of the function. When you differentiate a function, you apply rules like the power rule to each term.

• The goal is to transform each expression into its derivative form.
• This process is essential for understanding how functions behave differently over intervals.

Using differentiation, we can solve for the first derivative \(\frac{dy}{dx}\) to understand the slope of the tangent line at any point on the curve. Taking this further, by differentiating a second time, you find the second derivative \(\frac{d^2y}{dx^2}\), which tells us about the concavity of the function.

Simplifying Functions for Easier Differentiation

Before differentiating, it's often beneficial to simplify a function as much as possible. Simplifying makes it easier to apply differentiation rules without confusion. For the function \(y = \frac{x^3 + 7}{x}\), simplify by dividing each term by \(x\):

• Rewrite as \(x^2 + 7x^{-1}\).
• This transformed the division into addition of terms with simple exponents.

Once simplified, you can readily apply the power rule to each term. The simplification step may seem small, but it is crucial for reducing errors and ensuring that differentiation is as straightforward as possible. By breaking down the function, you engage with each part separately, making the problem more manageable.