Problem 36 - Calculus an Applied Approach

Question

Find \(f^{\prime}(x)\) $$ f(x)=x^{2}-3 x-3 x^{-2}+5 x^{-3} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = x^{2}-3 x-3 x^{-2}+5 x^{-3}\) is \(f'(x) = 2x - 3 + 6x^{-3} + 15x^{-4}\).

Step 1: Identify Every Term of the Function

Consider the function \(f(x) = x^{2}-3 x-3 x^{-2}+5 x^{-3}\). Here, there are four terms that are \(x^{2}\), \(-3x\), \(-3x^{-2}\), and \(5x^{-3}\).

Step 2: Apply Power Rule for Derivation to Each Term

Apply the power rule as \(\frac{d}{dx}(x^n) = n*x^{n-1}\). \nThe derivative of \(x^{2}\) is \(\frac{d}{dx}(x^2) = 2*x^{2-1} = 2x\).\nThe derivative of \(-3x\) is \(\frac{d}{dx}(-3x) = -3*x^{1-1} = -3\).\nThe derivative of \(-3x^{-2}\) is \(\frac{d}{dx}(-3x^{-2}) = -3*(-2)*x^{-2-1} = 6x^{-3}\).\nThe derivative of \(5x^{-3}\) is \(\frac{d}{dx}(5x^{-3}) = 5*(-3)*x^{-3-1} = 15x^{-4}\).

Step 3: Combine All Derivatives

Combine the derivatives found in the previous step. The combined derivative of \(f(x)\) will be \(f'(x) = 2x - 3 + 6x^{-3} + 15x^{-4}\).

Key Concepts

Power Rule

The Power Rule is a fundamental formula in calculus used for differentiation, one of the core concepts in analyzing functions. The Power Rule is applicable to any polynomial term of the form \( x^n \). When you differentiate \( x^n \) with respect to \( x \), you apply the rule

• \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \)

Here, \( n \) is a real number which represents the exponent of the variable \( x \). The rule instructs you to multiply the original exponent \( n \) by the coefficient of the term and then reduce the exponent by one.
This makes it very straightforward to differentiate terms with powers. For example, differentiating \( x^3 \) gives \( 3x^2 \) because you multiply the exponent (3) by the coefficient (1) and reduce the exponent by one (from 3 to 2).
The Power Rule is incredibly useful because it's easily applicable to any power, positive or negative, and even fractions.

Basic Differentiation

Basic differentiation involves finding the rate at which a function changes, which is represented by its derivative. A derivative tells you how a function behaves at any point along its curve.

• The derivative measures the slope of the function at a specific point.
• It helps in determining the function’s rate of change.
• Key operations include the application of rules like the Power Rule, product rule, and chain rule.

To perform basic differentiation, start with recognizing individual terms within a function. Each term is differentiated separately using simple rules, mainly focusing on the Power Rule for polynomial terms.
For instance, take the polynomial function \( f(x) = x^2 - 3x \). You differentiate by using rules such as the power rule for \( x^2 \) and the constant multiple rule for \(-3x\). This gives the derivative \( f'(x) = 2x - 3 \).
Basic differentiation is essential for experiencing how functions transition and change their shape, demonstrating their growth, decay, or overall behavior.

Polynomial Functions

A polynomial function is a mathematical expression consisting of variables (unknowns) raised to whole number exponents and multiplied by coefficients. They take the form \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]

• Here, \( a_n, a_{n-1}, \ldots, a_0 \) are constants known as coefficients, and \( n \) is a non-negative integer representing the degree of the polynomial.
• The highest exponent signifies the polynomial's degree, representing the function’s leading term.
• Polynomials can range from simple linear functions to more complex quadratic or cubic functions and beyond.

The properties of polynomial functions make them continuously differentiable, meaning they can be differentiated at any point in their domain.
This smoothness enables their use in applications needing precise modeling, such as physics for motion trajectories or economics for trend predictions.
To differentiate a polynomial like \( f(x) = x^2 - 3x - 3x^{-2} + 5x^{-3} \), the terms are differentiated separately, respecting their power, using the Power Rule.
The derivative then captures the rate of change for the entire polynomial, forming a new function for analyzing the original function's behavior.