Problem 11 - University Calculus: Early Transcendentals

Question

Find the indicated derivatives. $$\frac{d p}{d q}\( if \)p=q^{3 / 2}$$

Step-by-Step Solution

Answer

The derivative \( \frac{d p}{d q} \) is \( \frac{3}{2} \sqrt{q} \).

Step 1: Understand the Problem

We need to find the derivative of the function \( p = q^{3/2} \) with respect to \( q \). This derivative is represented by \( \frac{d p}{d q} \).

Step 2: Recall the Power Rule

The power rule states that if \( p = q^n \), then the derivative \( \frac{d p}{d q} = n \cdot q^{n-1} \). Here, \( n = \frac{3}{2} \).

Step 3: Apply the Power Rule

Apply the power rule to the function. Substitute \( n = \frac{3}{2} \) into the power rule formula to find the derivative: \( \frac{d p}{d q} = \frac{3}{2} \cdot q^{\frac{3}{2} - 1} = \frac{3}{2} \cdot q^{\frac{1}{2}} \).

Step 4: Simplify the Expression

Simplify \( q^{\frac{1}{2}} \) to its simplest form. \( q^{\frac{1}{2}} \) is the square root of \( q \), so the expression becomes \( \frac{3}{2} \sqrt{q} \).

Key Concepts

Power Rule

The power rule is a foundational concept in calculus used to find the derivative of power functions, helping to measure the rate of change of a function. This rule states that for any function of the form \( p = q^n \) where \( n \) is any real number, the derivative of \( p \) with respect to \( q \) is \( \frac{d p}{d q} = n \cdot q^{n-1} \).

In simpler terms, this formula allows you to multiply the exponent by the base raised to a power that is one less than the original exponent.

• For example, if \( n = 3 \), the derivative of \( q^3 \) is \( 3q^{3-1} = 3q^2 \).
• For a fractional exponent, such as \( n = \frac{3}{2} \), the derivative becomes \( \frac{3}{2}q^{\frac{1}{2}} \), a key step in solving our original exercise.

This rule simplifies the process of differentiation, making it easier to work with complex equations by standardizing the way derivatives of power functions are calculated. It’s handy because it transforms a potentially complicated operation into a straightforward calculation that is easy to repeat with confidence.

Differentiation

Differentiation is the process of finding the derivative of a function. It's the mathematical way to find how a function changes at any given point, essentially measuring the function's rate of change.
Differentiation involves a set of rules and techniques, among which the power rule is one of the most commonly used for functions expressed as powers of a variable.

Differentiation is crucial for:

• Analyzing and understanding the behavior of functions, especially in terms of rate changes.
• Simplifying complex problems, allowing us to make predictions about data trends.
• Finding critical points in a graph, like where functions increase, decrease, or when they come to a halt (known as maxima and minima).

In our specific case of \( p = q^{3/2} \), differentiating gives us \( \frac{d p}{d q} = \frac{3}{2} \sqrt{q} \), which tells us the rate at which \( p \) changes as \( q \) changes.

Calculus

Calculus is a branch of mathematics that studies continuous change, through derivatives and integrals. While derivatives help us understand the rate of change, integrals help us find the total change over an interval.
Derivative operations form the core of differential calculus, and pivotal rules like the power rule make calculus more approachable.

Calculus lets us:

• Describe physical phenomena such as motion, electricity flow, and various scientific trends.
• Optimize solutions to real-world problems by identifying maximum and minimum values.
• Model situations in physics, economics, engineering, and more.

By applying principles of calculus, such as finding the derivative using the power rule, we gain insights into how quantities dynamically interact. This particular exercise showcases how even fractional powers can be handled with standard techniques, making advanced concepts accessible and usable in practical situations.