Problem 8 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=-2 x^{3}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = -2x^3\) is \(f'(x) = -6x^2\).

Step 1: Identify the power rule

The power rule states that for a function of the form \(f(x) = ax^n\), its derivative is given by \(f'(x) = n \cdot ax^{n - 1}\). In this case, we have the function \(f(x) = -2x^3\), so our values for a and n are: \(a = -2\) \(n = 3\)

Step 2: Apply the power rule

Now we will apply the power rule using the values we identified in Step 1. The derivative of the given function is: \(f'(x) = n \cdot a \cdot x^{n - 1}\) Substitute the values of a and n: \(f'(x) = (3)(-2) \cdot x^{3 - 1}\)

Step 3: Simplify the expression

Now we will simplify the expression to obtain the final derivative of the given function: \(f'(x) = -6x^{2}\) The derivative of the function \(f(x) = -2x^3\) is \(f'(x) = -6x^2\).

Key Concepts

Power Rule

The power rule is one of the most fundamental rules in calculus for finding derivatives. It provides a simple and efficient way to differentiate functions of the form \(f(x) = ax^n\). When you apply the power rule, you multiply the exponent \(n\) by the coefficient \(a\) and then decrease the exponent by one. This can be concisely written as:

• Derivative formula: \(f'(x) = n \cdot ax^{n - 1}\)
• Example: For \(f(x) = -2x^3\), we get \(f'(x) = (3)(-2)x^{2}\)

Using the power rule simplifies the process significantly, making it an indispensable tool for differentiation. Knowing how to apply this rule will enable you to manage polynomial functions effectively and tackle more advanced calculus problems as you progress.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate at which a function is changing at any given point. In more practical terms, it tells you the slope of the tangent line at any point on the curve of the function.

• The derivative of a function \(f(x)\) is denoted as \(f'(x)\) or \(\frac{df}{dx}\).
• In our exercise, differentiation was used to find \(f'(x) = -6x^2\), the derivative of \(f(x) = -2x^3\).

Differentiation is central to calculus and extends beyond simple polynomials to include exponential, logarithmic, and trigonometric functions. Understanding differentiation is crucial as it lays the groundwork for concepts like optimization and motion analysis.

Mathematical Functions

Mathematical functions are expressions that describe a relationship between inputs and outputs. In calculus, functions often represent how one quantity changes in relation to another.

• Functions can be linear, quadratic, polynomial, exponential, and more.
• Each type of function has specific rules for differentiation and integration.
• The function \(f(x) = -2x^3\) in our example is a cubic polynomial.

Understanding the properties of different types of functions is key to successfully applying derivative rules. Recognizing the structure of functions allows you to select the correct differentiation techniques, ensuring accurate results in calculus applications.