Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=-x^{3}+2 x^{2}-6\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = -x^3 + 2x^2 - 6\) is \(f'(x) = -3x^{-4} + 4x\).

Step 1: Identify the terms of the function

The function \(f(x)\) has three terms: \(-x^3\), \(2x^2\), and \(-6\). We will apply the power rule to each of these terms.

Step 2: Apply the power rule to the first term (-x³)

The first term is \(-x^3\). Let's find its derivative using the power rule, \(f'(x) = n \cdot x^{n-1}\). Here, \(n=-3\). So, we get the derivative as follows: \(-3x^{(-3-1)}\)= \(-3x^{-4}\)

Step 3: Apply the power rule to the second term (2x²)

The second term is \(2x^2\). We can find its derivative using the power rule, \(f'(x) = n \cdot x^{n-1}\). Here, \(n=2\). So, we get the derivative as follows: \(2(2x^{(2-1)})\)= \(4x^{1}\) or simply \(4x\).

Step 4: Apply the power rule to the third term (-6)

The third term is \(-6\). Since this term does not have any x variable, its derivative becomes zero.

Step 5: Combine the derivatives

Now, let's combine the derivatives of the three terms. As a result: \[f'(x) = -3x^{-4} + 4x + 0\]

Step 6: Simplify the result

The simplified form of the derivative is as follows: \[f'(x) = -3x^{-4} + 4x\] So, the derivative of the function \(f(x) = -x^3 + 2x^2 - 6\) is: \[f'(x) = -3x^{-4} + 4x\]

Key Concepts

Power Rule

The Power Rule is one of the most foundational concepts in calculus that makes finding derivatives of polynomial functions much easier. Simply put, it provides a quick and efficient method to differentiate terms that are in the form of a power of x. For any function term like \( ax^n \), where \( a \) is a constant and \( n \) is a real number, the Power Rule states:

• Take the exponent \( n \) and multiply it by the coefficient \( a \), resulting in \( n imes ax^{n-1} \).
• Reduce the exponent by 1, which produces \( x^{n-1} \).

This simple operation can be applied to each term in a polynomial function, streamlining the process. In our exercise, the function \( f(x) = -x^3 + 2x^2 - 6 \) contains terms that are perfect candidates for the Power Rule. Here:

• The term \( -x^3 \) becomes \( -3x^{3-1} = -3x^2 \).
• The term \( 2x^2 \) becomes \( 2 \times 2x^{2-1} = 4x \).
• The constant \( -6 \) becomes 0 since constants derive to zero.

Derivative of Polynomial Functions

Finding the derivative of a polynomial function involves taking the derivative of each term individually and then summing those derivatives. Polynomial functions are expressions involving only sums and constant multiples of powers of the variable \( x \). Following the derivative rules:

• Each term \( ax^n \) uses the Power Rule to find \( nax^{n-1} \).
• Constant terms are differentiated to zero because they do not change as \( x \) changes.

Let us fully consider our function \( f(x) = -x^3 + 2x^2 - 6 \):

• The derivative of \( -x^3 \) is \( -3x^2 \), utilizing the power rule where the power reduces by one and multiplies by the coefficient.
• The derivative of \( 2x^2 \) similarly is \( 4x \).
• Finally, the constant \( -6 \) becomes zero. This illustrates why constant forces do not affect the rate of change.

Combining these, the derivative function is \( f'(x) = -3x^2 + 4x \). This result explains the slope of the tangent to the curve at any value of \( x \).

Mathematics Education

Understanding differentiation and the Power Rule is crucial in developing essential skills in calculus. Learning how to differentiate fosters an ability to analyze complex functions, providing insight into how changes in variables influence outcomes. Differentiation is not only fundamental to higher mathematics but also serves real-world applications in fields like economics, physics, engineering, and computer science. Students often benefit from seeing the step-by-step approach:

• Identify each term in the function separately.
• Apply known rules systematically, like the Power Rule for different terms.
• Combine the results logically to get the derivative.

By mastering these techniques, students gain confidence in their problem-solving abilities, equipping them with skills that extend beyond the classroom. The Power Rule, while simple, opens the door to deeper exploration of derivative functions and their applications, making it a valuable part of the mathematics education curriculum.