Problem 20 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(r)=\frac{4}{3} \pi r^{3} $$

Step-by-Step Solution

Answer

The derivative is \( f'(r) = 4\pi r^2 \).

Step 1: Identify the Derivative Rule

The function given is a power function with a constant coefficient. To find the derivative, we will use the power rule. For a function of the form \( f(x) = ax^n \), the derivative is \( f'(x) = anx^{n-1} \).

Step 2: Apply the Power Rule

In the function \( f(r) = \frac{4}{3} \pi r^3 \), the coefficient \( a \) is \( \frac{4}{3} \pi \) and the power \( n \) is 3. Applying the power rule gives: \[ f'(r) = 3 \times \frac{4}{3} \pi r^{3-1} \]

Step 3: Simplify the Expression

Simplify the derivative found in the previous step: \[ f'(r) = 3 \times \frac{4}{3} \pi r^2 = 4 \pi r^2 \] Therefore, the derivative of the function \( f(r) = \frac{4}{3} \pi r^3 \) is \( f'(r) = 4 \pi r^2 \).

Key Concepts

The Power Rule in Calculus

The power rule is a fundamental tool when working with derivatives of polynomial functions. It is the go-to method for finding the derivative of functions that are expressed as a variable raised to a power. The basic form of the power rule states that for any function of the form \( f(x) = ax^n \), the derivative \( f'(x) \) is \( anx^{n-1} \).

• This means you multiply the current power of the variable by the coefficient in front.
• Then, reduce the power of the variable by one.
• The power rule simplifies the process of differentiation, especially in polynomials where you can apply it term by term.

In the problem provided, the function \( f(r) = \frac{4}{3} \pi r^3 \) perfectly fits the power rule. By recognizing the format \( r^3 \), you can quickly apply the rule to find the derivative \( f'(r) \). Differentiating becomes straightforward and essential for analyzing function behavior.

Handling Constants in Derivatives

Constants play a significant role in the process of differentiation. When differentiating a function with constants, such as \( \frac{4}{3} \pi \) in the provided exercise, these constants are preserved during the derivative process.

• To manage constants when applying the power rule, treat them as multipliers outside of the derivative operation.
• The constant multiplier remains attached to the derivative of the variable term.
• This is crucial when simplifying the derivative at the end of your calculations.

In the given function \( f(r) = \frac{4}{3} \pi r^3 \), when finding the derivative, the constant \( \frac{4}{3} \pi \) stays in place. You only apply the power rule to the term \( r^3 \), resulting in the derivative \( f'(r) = 3 \times \frac{4}{3} \pi r^{3-1} \). Simplification then follows.

Simplifying Expressions in Derivatives

After finding a derivative using the power rule, simplifying the expression is often the final step. Simplification makes the derivative easier to understand and work with, especially in further calculations or applications.

• Begin by performing any arithmetic calculations, such as multiplying coefficients.
• Next, rewrite the expression in a more compact and understandable form.
• Simplifying helps in identifying the nature and features of the graph of the derived function.

In our exercise, the derivative at the point of simplification is \( 3 \times \frac{4}{3} \pi r^2 \). Multiplying the coefficients \( 3 \) and \( \frac{4}{3} \) simplifies to \( 4 \). Thus, the simplified derivative is \( f'(r) = 4 \pi r^2 \). This clear expression is vital for interpretations, such as understanding the rate of change of the function as \( r \) changes.