Problem 43 - Calculus and its applications

Question

Find \(f^{\prime}(x)\) for the given function. \(f(x)=x^{5}\) (See Exercise 50 in Section 1.3.)

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = 5x^4 \).

Step 1: Identify the Differentiation Rule

We need to find the derivative of the function \( f(x) = x^5 \). The appropriate rule for this purpose is the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \).

Step 2: Apply the Power Rule

Using the power rule, we differentiate \( x^5 \). According to the rule, we bring down the exponent as a coefficient and subtract one from the exponent: \( 5x^{5-1} = 5x^4 \).

Step 3: Write the Final Answer

The derivative \( f^{\prime}(x) \) of the function \( f(x) = x^5 \) is \( 5x^4 \).

Key Concepts

Power Rule

The Power Rule is an essential tool in differentiation. It's extremely helpful for swiftly finding the derivative of functions where variables are raised to a power. To apply the Power Rule in calculus, you look at any function of the form \( x^n \), where \( n \) is any real number. The Power Rule states that if you differentiate \( x^n \), the result is \( nx^{n-1} \).

• For example, if you have \( f(x) = x^5 \), the exponent 5 becomes the coefficient.
• You then subtract 1 from the original exponent to give \( n-1 \).
• The derivative thus becomes \( f'(x) = 5x^{5-1} = 5x^4 \).

This rule is quick and very effective, especially when dealing with polynomial functions, allowing us to find slopes or rates of change without much hassle.

Derivative

The derivative of a function is a fundamental concept in calculus. It measures how a function's output changes as its input changes; essentially, it gives us the rate of change or the slope of the function's graph at any given point. When you have a function \( f(x) \), the derivative, denoted \( f'(x) \), represents this rate of change.

• The process of finding a derivative is called differentiation.
• For a function like \( f(x) = x^5 \), the derivative \( f'(x) = 5x^4 \) tells us how steep the curve is at any point \( x \).
• This is crucial for understanding motion, growth, and any situation where things are constantly changing.

Derivatives are not just limited to simple power functions; they apply widely across different functions and scenarios in calculus.

Calculus

Calculus is a branch of mathematics focusing on change and motion. It splits into two main parts: differentiation and integration, both of which are interconnected. While differentiation deals with rates of change and slopes of curves, integration is concerned with accumulation and areas under curves.

• Calculus provides the tools to model and analyze dynamic systems.
• In physics, it helps in understanding motion, forces, and energy.
• In economics, it can analyze changes in cost, revenue, and profit over time.
• In biology, it models population growth and decay.

Understanding calculus opens up the ability to tackle complex problems and provides insights into the world by describing how things change over time or space.