Problem 47 - University Calculus: Early Transcendentals

Question

Find the first and second derivatives of the functions. $$r=\frac{(\theta-1)\left(\theta^{2}+\theta+1\right)}{\theta^{3}}$$

Step-by-Step Solution

Answer

First derivative: \( \frac{3}{\theta^4} \); Second derivative: \( \frac{-12}{\theta^5} \).

Step 1: Rewrite the Function in an Easier Form

Let's rewrite the given function for easier differentiation. The function given is: \[ r = \frac{(\theta - 1)(\theta^2 + \theta + 1)}{\theta^3} \]We can split this into two separate functions and simplify the expression by multiplying the terms in the numerator:\[ r = \frac{\theta^3 - \theta^2 + \theta^2 - \theta + \theta - 1}{\theta^3} = \frac{\theta^3 - 1}{\theta^3} \]Divide each term separately:\[ r = 1 - \frac{1}{\theta^3} \]

Step 2: Find the First Derivative

We will now find the first derivative \( \frac{dr}{d\theta} \) of the simplified function \( r = 1 - \theta^{-3} \).The derivative of a constant (\(1\)) is zero, and the derivative of \( -\theta^{-3} \) using power rule \((d/dx)x^n = nx^{n-1})\) is:\[ \frac{dr}{d\theta} = 0 - (-3)\theta^{-3-1} = 3\theta^{-4} \]Thus:\[ \frac{dr}{d\theta} = \frac{3}{\theta^4} \]

Step 3: Find the Second Derivative

Now find the second derivative \( \frac{d^2r}{d\theta^2} \) from the first derivative \( \frac{dr}{d\theta} = 3\theta^{-4} \).Applying the power rule \((n\theta^{n-1}))\) again:\[ \frac{d^2r}{d\theta^2} = 3(-4)\theta^{-4-1} = -12\theta^{-5} \]Thus:\[ \frac{d^2r}{d\theta^2} = \frac{-12}{\theta^5} \]

Key Concepts

First Derivative

The first derivative is a cornerstone in calculus. It helps us understand how a function changes as its input changes.
For the function described here, finding the first derivative means determining the rate at which the function \( r \) changes with respect to \( \theta \).

• Start by simplifying the original function, \( r = \frac{(\theta - 1)(\theta^2 + \theta + 1)}{\theta^3} \), to a more manageable form.
  In this case, breaking down the multiplication in the numerator and simplifying it resulted in a function that is easier to derive: \( r = 1 - \theta^{-3} \).
• The derivative of \( r \) gives us \( \frac{dr}{d\theta} \). In the simplified form, one term is a constant (\(1\)), and the other is \( -\theta^{-3} \).
  Applying the power rule, where \( d/dx(x^n) = nx^{n-1} \), the derivative of \( -\theta^{-3} \) is \( +3\theta^{-4} \).
  This means: \[ \frac{dr}{d\theta} = \frac{3}{\theta^4} \]

By finding the first derivative, we learn about the instantaneous rate of change of our function at any point \( \theta \). This is vital for understanding behaviors like increasing or decreasing trends.

Second Derivative

The second derivative provides even deeper insights into a function's behavior. Once you have the first derivative, the second derivative shows how the rate of change itself is changing.
Consider the first derivative we found: \( \frac{dr}{d\theta} = 3\theta^{-4} \).
To find the second derivative, we apply the power rule again:

• Take the derivative of the first derivative using \( n\theta^{n-1} \).
  Here, \( 3\theta^{-4} \) becomes \( -12\theta^{-5} \).
• The result, \( \frac{d^2r}{d\theta^2} = \frac{-12}{\theta^5} \), helps indicate concavity.
  Concavity tells us if the function is curving upwards or downwards at a given \( \theta \).

The second derivative can reveal points of inflection and how the function's curvature changes, offering advanced insights into the function's dynamics.

Power Rule

The power rule is one of the most fundamental rules for differentiation. It's powerful for finding derivatives efficiently, especially in polynomial functions.
In simple terms, if you have a function in the form \( x^n \), the derivative is \( nx^{n-1} \).

• This rule makes differentiating straightforward by reducing the power by one and multiplying the term by the original power.
• In our exercise, use the power rule to derive both the first and second derivatives.
  Starting with \( r = 1 - \theta^{-3} \), differentiating \( -\theta^{-3} \) gives you \( +3\theta^{-4} \) using the power rule.
  Next, \( 3\theta^{-4} \) is differentiated again to obtain \( -12\theta^{-5} \).

The power rule simplifies the process of finding derivatives, allowing for quick and error-free calculations, which are crucial for more complex problems.