Problem 4 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=x^{7}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^7\) using the power rule of differentiation is \(f'(x) = 7x^6\).

Step 1: Identify the rule needed for differentiation

For this function, we will apply the power rule of differentiation, which states that for any function in the form \(f(x) = x^n\), where \(n\) is a constant, the derivative is given by \(f'(x) = nx^{n-1}\).

Step 2: Apply the power rule to the given function

We have the function \(f(x) = x^7\) and we want to find its derivative, \(f'(x)\). According to the power rule, we will multiply the current exponent (\(7\)) by the variable (\(x\)), and then subtract \(1\) from the exponent, resulting in: \[f'(x) = 7x^{7-1}\]

Step 3: Simplify the expression for the derivative

Now simplify the expression by calculating the new exponent: \[f'(x) = 7x^{6}\] So, the derivative of the function \(f(x) = x^7\) is \(f'(x) = 7x^6\).

Key Concepts

Understanding the Power Rule

When learning about differentiation in calculus, the power rule is a crucial concept. It simplifies the process of finding the derivative of functions, particularly those expressed as polynomial expressions. The power rule states that if you have a function in the form of \(f(x) = x^n\), where \(n\) is a constant, the derivative \(f'(x)\) is \(nx^{n-1}\). This means you multiply the exponent by the base, which is the variable, then subtract one from the exponent.

For example, if you have \(f(x) = x^7\), you apply the power rule by multiplying the 7 by \(x\) and reducing the exponent by one, resulting in the derivative \(f'(x) = 7x^6\).

• Write the original exponent next to the variable as a coefficient.
• Decrease the original exponent by 1 to get the new exponent.

This method is not only simple but also powerful for quickly differentiating polynomials and can be extended to negative and fractional exponents as well.

The Derivative and its Importance

In calculus, the derivative represents the rate at which a function is changing at any given point. Essentially, it tells us how the function's value changes with respect to changes in the variable. Finding derivatives is a foundational tool because it allows us to understand and predict the behavior of functions.

The derivative is significant for several reasons:

• Helps determine the slope of a function at any specific point.
• Aids in solving problems involving rates of change, such as speed and acceleration.
• Supports the analysis of complex systems in physics, engineering, and other sciences.

By computing derivatives, one can also identify crucial points on graphs such as maxima, minima, and inflection points, leading to a better comprehension of how functions behave and interact.

The Role of Calculus in Mathematics

Calculus is the branch of mathematics that deals with continuous change, and it plays a critical role in various fields such as physics, engineering, economics, and biology. It divides into two major parts: differentiation and integration.

Differentiation focuses on finding the derivative, which indicates an instantaneous rate of change, as opposed to a simple rate of change. This is essential for solving real-world problems that involve dynamic systems.

• Enables the modeling of systems where variables are in constant flux.
• Allows for the analysis of functions to determine growth, trends, and other behaviors.

Furthermore, calculus helps pave the way for more advanced mathematical studies and applications, enabling innovation and development across diverse scientific disciplines. Understanding calculus and its components, such as the power rule and derivatives, is essential to solving complex problems in both academia and industry.