Problem 8 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{1}{3} x^{9} $$

Step-by-Step Solution

Answer

The derivative of the function is \(f'(x) = 3x^8\).

Step 1: Identify the Formula for the Power Rule

To find the derivative of a function of the form \(f(x) = ax^n\), we apply the power rule. The power rule formula is given by \(f'(x) = n imes ax^{n-1}\), where \(a\) is a constant, \(n\) is the exponent, and \(f'(x)\) is the derivative of \(f(x)\).

Step 2: Apply the Power Rule

In our function \(f(x) = \frac{1}{3}x^9\), we identify \(a = \frac{1}{3}\) and \(n = 9\). Using the power rule, the derivative \(f'(x)\) is calculated as \(9 \times \frac{1}{3} x^{9-1}\).

Step 3: Simplify the Derivative

Simplify the derivative expression from Step 2. Calculate the product \(9 \times \frac{1}{3}\), which simplifies to \(3\). Therefore, \(f'(x) = 3x^{8}\), since \(9-1 = 8\).

Key Concepts

Power Rule

The power rule is an essential tool in calculus used for differentiation. It is especially useful when dealing with polynomial functions.
The power rule makes finding derivatives of functions in the form of \( f(x) = ax^n \) straightforward. Here are the main points to remember about the power rule:

• Identify the coefficient \( a \) and the exponent \( n \) in your polynomial function.
• The rule states that the derivative of \( f(x) = ax^n \) is given by \( f'(x) = n \times ax^{n-1} \).
• Apply the rule by multiplying the coefficient \( a \) by the exponent \( n \), and reduce the exponent by one.
• This process simplifies a potentially complex calculation.

In the exercise, for the function \( f(x) = \frac{1}{3} x^9 \), we used \( a = \frac{1}{3} \) and \( n = 9 \). Applying the power rule gives \( f'(x) = 9 \times \frac{1}{3} x^{8} \), resulting in \( 3x^8 \) after simplification.

Differentiation

Differentiation is the process of finding the derivative, which is a measure of how a function changes as its input changes. The derivative tells us the rate of change or the slope of the function at any given point.

• It's a foundational concept in calculus, used to understand how functions behave.
• The basic approach to differentiation involves applying rules like the power rule to compute the derivative efficiently.
• For many functions, differentiation can show how gradual or rapid changes occur, which is critical in fields like physics, engineering, and economics.
• A derivative can tell us whether a function is increasing or decreasing at a certain point.

In our given example, by differentiating \( f(x) = \frac{1}{3} x^9 \), we determined \( f'(x) \), which tells us about the rate at which \( f(x) \) changes with respect to \( x \). The derivative, \( 3x^8 \), indicates this rate.

Calculus

Calculus is the branch of mathematics that studies continuous change. It is divided mainly into two parts: differentiation and integration. Both are essential for understanding the dynamics of different systems.

• Calculus enables us to model and analyze systems and functions in a precise way.
• It provides the necessary tools for finding changes (differentiation) and total amounts accumulated through change (integration).
• Applications of calculus range from predicting the movement of planets to optimizing business strategies.
• By learning the fundamentals like the power rule and differentiation, you begin to unlock many complex concepts in calculus.

Understanding the power rule and differentiation as applied in the exercise is a step towards grasping broader concepts in calculus, leading to significant real-world applications.