Problem 8 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, find the first and second derivative of the function defined by the given equation. $$ g(r)=\sqrt{r}+\frac{1}{\sqrt{r}} $$

Step-by-Step Solution

Answer

First derivative: \( g'(r) = \frac{1}{2}r^{-1/2} - \frac{1}{2}r^{-3/2} \). Second derivative: \( -\frac{1}{4}r^{-3/2} + \frac{3}{4}r^{-5/2} \).

Step 1: Rewrite the Function

Rewrite the given function in terms of exponents for easier differentiation. The function is given as:\[ g(r) = \sqrt{r} + \frac{1}{\sqrt{r}} \]Rewrite this as:\[ g(r) = r^{1/2} + r^{-1/2} \]

Step 2: Find the First Derivative

Differentiate the function \( g(r) = r^{1/2} + r^{-1/2} \) using the power rule. The power rule states that the derivative of \( r^n \) is \( n \cdot r^{n-1} \). So, applying this:\[ g'(r) = \frac{1}{2} r^{-1/2} - \frac{1}{2} r^{-3/2} \]This simplifies to:\[ g'(r) = \frac{1}{2\sqrt{r}} - \frac{1}{2r^{3/2}} \]

Step 3: Simplify the First Derivative

Combine the terms to find a common denominator if needed:\[ g'(r) = \frac{1}{2\sqrt{r}} - \frac{1}{2r^{3/2}} \]Alternatively:\[ g'(r) = \frac{1}{2}r^{-1/2} - \frac{1}{2}r^{-3/2} \]

Step 4: Find the Second Derivative

Differentiate the first derivative \( g'(r) = \frac{1}{2}r^{-1/2} - \frac{1}{2}r^{-3/2} \) using the power rule again.The power rule gives us:\[ g''(r) = \frac{1}{2} \left(-\frac{1}{2}r^{-3/2}\right) - \frac{1}{2} \left(-\frac{3}{2}r^{-5/2}\right) \]This simplifies to:\[ g''(r) = -\frac{1}{4}r^{-3/2} + \frac{3}{4}r^{-5/2} \]

Step 5: Simplify the Second Derivative

Combine the terms if needed:\[ g''(r) = -\frac{1}{4r^{3/2}} + \frac{3}{4r^{5/2}} \]Alternatively:\[ g''(r) = -\frac{1}{4}r^{-3/2} + \frac{3}{4}r^{-5/2} \]

Key Concepts

First Derivative

To understand the concept of the first derivative, let's start with its definition. The first derivative of a function gives us the rate at which the function's value changes with respect to its variable. In simpler terms, it tells us the slope of the function at any given point.

For example, consider the function provided in the exercise: \[ g(r) = \frac{1}{2\root{r}} - \frac{1}{2r^{3/2}} \]. The first derivative \( g'(r) \) is determined by applying the power rule for differentiation.

The power rule for differentiation states:

• If \( y = x^n \), then \( \frac{dy}{dx} = n x^{n-1} \).

This rule helps us find the rate of change for functions involving a variable to some power.
Following the steps provided in the solution, the differentiation of \( r^{1/2} + r^{-1/2} \) results in \( g'(r) = \frac{1}{2\root{r}} - \frac{1}{2r^{3/2}} \). This first derivative function describes the slope of the original function at each value of \( r \). This slope also indicates how steep or flat the function graph appears at particular points.

Second Derivative

The second derivative provides even more insights. It tells us how the rate of change of a function's slope is itself changing. Essentially, while the first derivative gives us the slope, the second derivative informs us about the curvature or concavity of the function's graph.

Continuing from our example function, we had already found the first derivative \( g'(r) = \frac{1}{2\root{r}} - \frac{1}{2r^{3/2}} \). The second derivative is found by differentiating \( g'(r) \) again using the power rule.

Using the power rule, we found:

• If \( g'(r) = r^n \), then \( g''(r) = n r^{n-1} \),

the second derivative results in \( g''(r) = -\frac{1}{4r^{3/2}} + \frac{3}{4r^{5/2}} \). This second derivative helps us understand whether our original function's graph is curving upwards or downwards at each point.
For instance:

• If the second derivative is positive, the function is concave up (like a cup).
• If the second derivative is negative, the function is concave down (like a frown).

Understanding both first and second derivatives can thus provide a comprehensive view of the function's behavior.

Power Rule for Differentiation

The power rule for differentiation is a fundamental technique in calculus for finding the derivative of functions of the form \( x^n \). It simplifies the process when dealing with polynomial expressions.

To recall, the power rule states: If you have a function \( y = x^n \), its derivative \( \frac{dy}{dx} = n x^{n-1} \). This rule is derived from the limit definition of a derivative and is key to efficiently differentiating functions with powers of the variable.

Let's look at how this rule was applied in our solved exercise: For the function \( g(r) = r^{1/2} + r^{-1/2} \), we applied the power rule to find both the first and second derivatives:

• First derivative: \( g'(r) = \frac{1}{2} r^{-1/2} - \frac{1}{2} r^{-3/2} \)
• Second derivative: \( g''(r) = -\frac{1}{4} r^{-3/2} + \frac{3}{4} r^{-5/2} \)

Each time, the power rule was used by bringing the exponent in front of the variable and subtracting one from the original exponent.
This rule not only simplifies differentiation but also makes it a consistent process for more complex functions.