Problem 45 - Calculus an Applied Approach

Question

Find \(f^{\prime}(x)\) $$ f(x)=\frac{4 x^{3}-3 x^{2}+2 x+5}{x^{2}} $$

Step-by-Step Solution

Answer

The derivative of \(f(x)= \frac{4 x^{3}-3 x^{2}+2 x+5}{x^{2}}\) is \(f^{\prime}(x)= \frac{4 x^{4}+2x^{2}-10x}{x^{4}}\).

Step 1: State the Quotient Rule

The Quotient Rule states that the derivative of \(\frac{u}{v}\) where both u and v are differentiable is given by \(\frac{u^{\prime}v-uv^{\prime}}{v^{2}}\). In this problem, \(u=4x^{3}-3x^{2}+2x+5\) and \(v = x^{2}\).

Step 2: Find the Derivatives of u and v

By application of the Power Rule, the derivative of \(u=4x^{3}-3x^{2}+2x+5\) is \(u^{\prime}=12 x^{2}-6x+2\). For \(v= x^{2}\), \(v^{\prime}=2x\)

Step 3: Apply the Quotient Rule

Substitute \(u\), \(u^{\prime}\), \(v\), and \(v^{\prime}\) into the Quotient Rule calculation. This results in \(f^{\prime}(x) = \frac{(12 x^{2}-6x+2)x^{2}-(4x^{3}-3x^{2}+2x+5)(2x)}{x^{4}}\).

Step 4: Simplify the Expression

On simplifying the numerator and the denominator, you get \(f^{\prime}(x) = \frac{12 x^{4}-6x^{3}+2x^{2}-8x^{4}+6x^{3}-4x^{2}-10x}{x^{4}}\). Further simplification gives \(f^{\prime}(x) = \frac{4 x^{4}+2x^{2}-10x}{x^{4}}\).

Key Concepts

Quotient Rule

When differentiating a function expressed as the division of two other functions, we apply the Quotient Rule. This is a fundamental tool in calculus for finding derivatives of rational expressions.

• The Quotient Rule states: if you have a function that is the quotient of two differentiable functions, say \( \frac{u}{v} \), then the derivative \( \frac{d}{dx}\left(\frac{u}{v}\right) \) is given by:

\[\frac{u^{\prime}v - uv^{\prime}}{v^{2}}\]Here, \( u \) is the numerator, \( v \) is the denominator, \( u' \) is the derivative of \( u \), and \( v' \) is the derivative of \( v \). The terms are substituted into the formula to find the derivative of the complete fraction. It's vital to ensure that both the numerator and denominator are differentiable for this rule to be applicable.

Power Rule

The Power Rule is a basic method used to find the derivative of polynomial expressions. It states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \).

• Consider the expression \( u = 4x^{3}-3x^{2}+2x+5 \):
• To differentiate each term: apply Power Rule individually:
• \( \frac{d}{dx}(4x^{3}) = 12x^{2} \)
• \( \frac{d}{dx}(-3x^{2}) = -6x \)
• \( \frac{d}{dx}(2x) = 2 \)
• \( \frac{d}{dx}(5) = 0 \) because 5 is a constant.
This results in \( u^{\prime} = 12x^{2} - 6x + 2 \). Applying the Power Rule simplifies finding derivatives, and is essential for manual differentiation, especially for each term within polynomial expressions.

Polynomial Function Differentiation

Differentiating polynomial functions is often straightforward, thanks to the simple nature of terms and application of the Power Rule. Polynomial functions like \( 4x^{3} - 3x^{2} + 2x + 5 \) are composed of terms where powers of \( x \) are whole numbers, making them differentiable using basic rules.

• Always identify each term in the polynomial.
• Apply differentiation term by term, using the Power Rule.

For complex expressions involving division like in this exercise, follow these steps:- Use the Quotient Rule for the overall function.- Decompose the function into manageable terms (numerator and denominator).- Differentiate each part applying the Power Rule or other necessary derivative rules as required.Mastering polynomial differentiation lays the groundwork for tackling more complex calculus problems. It's a crucial skill that builds efficiency and accuracy in solving mathematical computations.