Problem 34 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$V=\frac{4}{3} \pi r^{2} b$$

Step-by-Step Solution

Answer

The derivative is \( \frac{8}{3}\pi br \).

Step 1: Identify the variable and constants

In the given expression, \(V=\frac{4}{3}\pi r^{2} b\), \( r \) is the variable, and \( \frac{4}{3}\pi \) and \( b \) are constants. This indicates that we will differentiate with respect to \( r \).

Step 2: Use the Power Rule for Differentiation

The power rule states that if \( f(r) = r^n \), then \( f'(r) = nr^{n-1} \). Here, \( r^2 \) is in our function, so calculate its derivative \( \frac{d}{dr} r^2 = 2r \).

Step 3: Differentiate Using the Constant Multiple Rule

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. Therefore, \(\frac{d}{dr}(\frac{4}{3}\pi r^2 b) = \frac{4}{3}\pi b \frac{d}{dr}(r^2) \). Substitute \( \frac{d}{dr}(r^2) = 2r \) from Step 2.

Step 4: Simplify the Expression

Substituting the derivative calculated, we have \( \frac{d}{dr} V = \frac{4}{3}\pi b (2r) = \frac{8}{3}\pi br \). Therefore, the derivative of \( V \) with respect to \( r \) is \( \frac{8}{3}\pi br \).

Key Concepts

Power Rule

In calculus, the power rule is a fundamental technique for finding the derivative of functions in the form of a power of a variable. It applies when you have a term like \(r^n\), where \(r\) is a variable and \(n\) is a real number exponent.
To find the derivative, you use the power rule formula: if \(f(r) = r^n\), then the derivative \(f'(r) = n\cdot r^{n-1}\). This means you bring the exponent \(n\) in front of the term and reduce the exponent by one.

• Example: For \(r^2\), using the power rule gives \(2r^{2-1} = 2r\).
• It's simple and quick, helping you solve many differentiation problems swiftly.

This fundamental rule simplifies the process of finding how functions change, making it a go-to shortcut in differentiation.

Constant Multiple Rule

The constant multiple rule in differentiation deals with expressions where a constant is multiplied by a variable function. A constant is a number that doesn’t change, like \( \frac{4}{3} \) or \(\pi b\) in our expression. This rule states that the derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function.
For example, if we have a function \(g(r) = c \cdot f(r)\), where \(c\) is a constant, the derivative is \(g'(r) = c \cdot f'(r)\).

• In practice: the derivative of \(\frac{4}{3}\pi br^2\) with respect to \(r\) implies first finding the derivative of \(r^2\) and then multiplying by the constant \(\frac{4}{3}\pi b\).

This rule is essential for simplifying complex derivatives involving constants, making calculations manageable.

Differentiation

Differentiation is a core concept in calculus that involves finding the rate at which a function is changing at any given point. This process allows you to find the derivative of a function, which is essentially the slope of the function at a specific point or the rate of change.
Differentiation answers questions like: how does a quantity change as another quantity changes, and it determines the sensitivity to changes in input. By applying rules such as the power rule and constant multiple rule, differentiation becomes a structured and predictable process.

• It involves steps that can be methodically applied to obtain the derivative of various types of functions.
• Understanding differentiation deeply enhances your ability to solve real-world problems involving motion, growth, and change.

With these tools, you can uncover insights into the behavior and characteristics of different functions.

Variables and Constants

In terms of differentiation, variables and constants have distinct roles within mathematical expressions. Variables are symbols that can represent different values. For instance, in the function \(V=\frac{4}{3}\pi r^{2} b\), \(r\) is a variable because it can change. Constants, like \(\frac{4}{3}\), \(\pi\), and \(b\), are fixed numbers that do not change, simplifying calculations and providing stability to expressions.
When differentiating, it is crucial to identify which components of your equation are constant and which are variable, as that influences how differentiation rules are applied:

• Variable: Differentiation will act directly on variables using rules such as the power rule.
• Constant: Constants "stay out of the way" during differentiation, typically being multiplied after differentiation of variables.

Recognizing variables and constants empowers you to navigate calculus problems effectively.