Problem 11 - University Calculus: Early Transcendentals

Question

Find the first and second derivatives. $$r=\frac{1}{3 s^{2}}-\frac{5}{2 s}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Components

The function given is composed of two terms: \( \frac{1}{3s^2} \) and \( \frac{-5}{2s} \). We need to find the derivatives of each component separately.

Step 2: Rewrite the Function

Rewriting the function in terms of powers makes differentiation easier: \( r = \frac{1}{3}s^{-2} - \frac{5}{2}s^{-1} \).

Step 3: First Derivative of the First Term

Differentiate \( \frac{1}{3}s^{-2} \) using the power rule. The derivative is \( -\frac{2}{3}s^{-3} \).

Step 4: First Derivative of the Second Term

Differentiate \( \frac{5}{2}s^{-1} \) using the power rule. The derivative is \( \frac{5}{2}s^{-2} \).

Step 5: Combine for First Derivative

Combine the results from Steps 3 and 4 to get the first derivative: \( r' = -\frac{2}{3}s^{-3} + \frac{5}{2}s^{-2} \).

Step 6: Second Derivative of the First Term

Differentiate \( -\frac{2}{3}s^{-3} \) to get the second derivative of the first term: \( \frac{6}{3}s^{-4} = 2s^{-4} \).

Step 7: Second Derivative of the Second Term

Differentiate \( \frac{5}{2}s^{-2} \) to get the second derivative of the second term: \( -5s^{-3} \).

Step 8: Combine for Second Derivative

Combine the results from Steps 6 and 7 to find the second derivative: \( r'' = 2s^{-4} - 5s^{-3} \).

Key Concepts

Power rule

The power rule is one of the most commonly used techniques in differentiation, particularly when you are dealing with polynomial functions. It states that when you have a function of the form \( f(x) = ax^n \), the derivative \( f'(x) \) is given by multiplying \( a \) by \( n \) and then reducing the power of \( x \) by one. In other words:
\[ f'(x) = an x^{n-1} \]
This simple rule allows us to find derivatives quickly and efficiently.

• For example, if you have \( s^{-2} \), when differentiating, you bring down the exponent "-2" and multiply it with the coefficient, which is \( \frac{1}{3} \) in this case, resulting in \(-\frac{2}{3}s^{-3}\).
• Similarly, for \( s^{-1} \), bring down "-1" to multiply it with \(-\frac{5}{2}\), resulting in \( \frac{5}{2}s^{-2}\).

This technique is invaluable as it gives us the tools to quickly assess changes in functions wherever a simple power of \( x \) is involved. Remember that the coefficient on the function affects the outcome, but the process is identical regardless of the initial power.

Differentiation

Differentiation is the process of finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point. It is a fundamental tool in calculus used to study motion, rates of change, and the shape of graphs.
To differentiate a function like \( r = \frac{1}{3}s^{-2} - \frac{5}{2}s^{-1} \), we apply rules such as the power rule described above:

• First, break down the function into simpler parts that can be differentiated separately.
• Apply the power rule to each term in the expression.
• Combine the results to find the derivative of the entire function.

This step-wise approach ensures that even complex expressions can be handled efficiently, allowing one to calculate slopes and understand behavior of functions with precision.
In the given exercise, differentiation helps us understand exactly how the function \( r \) changes concerning \( s \), allowing us to pinpoint specific rates of transformation. Using differentiation, we can derive other useful calculations like the second derivative, which we discuss next.

Second derivative

The second derivative of a function is simply the derivative of its derivative. It provides information about the curvature or concavity of the original function. The first derivative informs us of the rate of change, while the second derivative tells us how this rate is changing.
In practical terms, a positive second derivative indicates that the original function is "concave up," resembling a bowl shape, while a negative second derivative indicates "concave down," like an upside-down bowl.

• In our problem, the function \( r' = -\frac{2}{3}s^{-3} + \frac{5}{2}s^{-2} \) was derived from the original function using the power rule.
• Taking the derivative of \( r' \) gives us \( r'' = 2s^{-4} - 5s^{-3} \), indicating how the rate of change itself changes as \( s \) varies.

Analyzing the second derivative is crucial for understanding the function’s behavior comprehensively, especially if you are evaluating maxima, minima, or inflection points where the character of change switches in a graph.