Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=\frac{x^{3}-4 x^{2}+3}{x}\)

Step-by-Step Solution

Answer

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

Step 1: Identify numerator and denominator

The given function is \(f(x)=\frac{x^{3}-4 x^{2}+3}{x}\). The numerator is the polynomial \(x^3 - 4x^2 + 3\), and the denominator is just \(x\).

Step 2: Apply the Quotient Rule

The Quotient Rule states that for any two differentiable functions \(u(x)\) and \(v(x)\), the derivative of their quotient is given by: \[\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}\] In our case, \(u(x) = x^3 - 4x^2 + 3\) and \(v(x) = x\). We need to find the derivatives of both \(u(x)\) and \(v(x)\) with respect to x: \[u'(x) = \frac{d}{dx}(x^3 - 4x^2 + 3) = 3x^2 - 8x\] \[v'(x) = \frac{d}{dx}(x) = 1\] Now, apply the Quotient Rule: \[f'(x) = \frac{(3x^2 - 8x)x - (x^3 - 4x^2 + 3)(1)}{x^2}\]

Step 3: Simplify the result

Let's now simplify the expression we obtained: \[f'(x) = \frac{3x^3 - 8x^2 - x^3 + 4x^2 - 3}{x^2}\] Combine similar terms: \[f'(x) = \frac{2x^3 - 4x^2 - 3}{x^2}\] The simplified expression for the derivative of the function is \(f'(x) = \frac{2x^3 - 4x^2 - 3}{x^2}\).

Key Concepts

Quotient Rule

When working with derivatives involving a division of two functions, the Quotient Rule offers a structured approach. The idea behind this rule is to find the derivative of a function expressed as a quotient, or division, of two differentiable functions. Given two functions, numerator \(u(x)\) and denominator \(v(x)\), the rule is expressed as:

• \[ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} \] This can be memorized as "\(u'v - uv'\) over \(v^2\)".

This method ensures that we account for how both the numerator and denominator change with respect to \(x\), capturing the entire dynamic of the function. It's crucial for handling cases where simple division might imply different behavior when variables are involved. Using this rule helps avoid errors when differentiating complex fractional forms. It efficiently distinguishes between shifts in both parts of the equation.

Polynomial Derivatives

Polynomials are mathematical expressions featuring variables raised to whole number powers and coefficients. Derivatives of polynomials are straightforward because of their predictable behavior.

• The standard rule for differentiating a term \(ax^n\) is \((n \times a)x^{n-1}\), where "a" is the coefficient, and "n" is the power.
• This means for the term \(x^3\), the derivative would be \(3x^2\).

This rule can be applied term by term, making polynomial functions excellent candidates for practice in differentiation. When a function like \(x^3 - 4x^2 + 3\) is differentiated with respect to \(x\), each term is dealt with individually:

• \(x^3\) becomes \(3x^2\)
• \(-4x^2\) becomes \(-8x\)
• And a constant \(3\) differentiates to \(0\)

This predictability and simplicity of polynomial derivatives make them a fundamental part of calculus.

Derivative Simplification

Once you have calculated the derivative of a function using the appropriate rules, it's important to simplify the result. Simplification helps in both understanding and further using the result.

• Simplifying derivatives involves combining like terms and reducing fractions when possible.
• This process often leads to a more elegant and manageable expression.

For example, the derivative \(f'(x) = \frac{3x^3 - 8x^2 - x^3 + 4x^2 - 3}{x^2}\) can be simplified:

• Combine terms like \(3x^3 - x^3\) to get \(2x^3\), and \(-8x^2 + 4x^2\) to get \(-4x^2\).
• You end up with \(\frac{2x^3 - 4x^2 - 3}{x^2}\).

Simplifying not only makes the derivative easier to interpret but also paves the way for further analysis, like finding critical points or inflection points.