Problem 115 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=\frac{4 x^{3}-2 x+1}{x^{2}}$$

Step-by-Step Solution

Answer

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

Step 1: Apply the Quotient Rule

The function \(f(x) = \frac{4x^3 - 2x + 1}{x^2}\) is a fraction, so the quotient rule should be used. The derivative of a quotient \(\frac{u(x)}{v(x)}\) is \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\) where \(u(x) = 4x^3 - 2x + 1\) and \(v(x) = x^2\).

Step 2: Find the Derivative of the Numerator

Differentiate the numerator, \(u(x) = 4x^3 - 2x + 1\). The derivative is \(u'(x) = 12x^2 - 2\).

Step 3: Find the Derivative of the Denominator

Differentiate the denominator, \(v(x) = x^2\). The derivative is \(v'(x) = 2x\).

Step 4: Apply the Derivatives in the Quotient Formula

Substitute \(u(x)\), \(u'(x)\), \(v(x)\), and \(v'(x)\) into the quotient rule: \[ f'(x) = \frac{(12x^2 - 2)x^2 - (4x^3 - 2x + 1)(2x)}{(x^2)^2} \].

Step 5: Simplify the Expression

Multiply out the terms in the numerator: \((12x^2 - 2)x^2 = 12x^4 - 2x^2\) and \((4x^3 - 2x + 1)(2x) = 8x^4 - 4x^2 + 2x\).Substituting these, the expression becomes:\[ f'(x) = \frac{12x^4 - 2x^2 - 8x^4 + 4x^2 - 2x}{x^4} \].

Step 6: Combine and Simplify Further

Combine like terms in the numerator:\((12x^4 - 8x^4) + (-2x^2 + 4x^2) - 2x = 4x^4 + 2x^2 - 2x\).So the expression becomes:\[ f'(x) = \frac{4x^4 + 2x^2 - 2x}{x^4} \].

Step 7: Simplify the Final Derivative

Rewrite each term such that the fraction simplifies:\[ f'(x) = \frac{4x^4}{x^4} + \frac{2x^2}{x^4} - \frac{2x}{x^4} = 4 + \frac{2}{x^2} - \frac{2}{x^3} \].

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental tool in calculus used to find the derivative of a function presented as a quotient of two other functions. When you have a function of the form \( \frac{u(x)}{v(x)} \), you cannot simply differentiate the numerator and the denominator separately like you do for products or sums. Instead, the derivative is calculated using the formula:

• \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \)

This formula arises from the product rule and chain rule applied together to a division scenario. Here, \( u(x) \) is the numerator and \( v(x) \) is the denominator. The Quotient Rule accurately accounts for the change in both the numerator and the denominator across an interval.
When applying the Quotient Rule, ensure:

• You correctly identify \( u(x) \) and \( v(x) \).
• Compute the derivatives \( u'(x) \) and \( v'(x) \) accurately.
• Substitute into the formula carefully.

Understanding this rule is essential for tackling any derivatives involving fractions.

Derivative of a Function

Finding the derivative of a function essentially reveals the rate at which it changes. Derivatives can be thought of as the slope of the tangent line to the function at any given point. For polynomial functions, you can apply the power rule to simply take the derivative:

• For a term like \( ax^n \), the derivative \( \frac{d}{dx}[ax^n] = nax^{n-1} \).

In the solution provided, we have differentiated both the numerator and the denominator separately. The numerator \( u(x) = 4x^3 - 2x + 1 \) was differentiated using the power rule to get \( u'(x) = 12x^2 - 2 \). Similarly, the denominator \( v(x) = x^2 \) gives us \( v'(x) = 2x \).
This differentiation gives us the components needed to use the Quotient Rule effectively. It is crucial to be meticulous during this step, as it forms the basis for correctly applying subsequent rules.

Simplifying Derivatives

Simplifying derivatives is the final and important step in solving problems that involve the derivative of a function. After applying the Quotient Rule, as shown in the exercise solution, you'll typically have a complex fraction that can be simplified.

• First, expand the terms by distributing changes through the products involved.
• Combine like terms. This means collecting terms with the same power of \(x\).
• Perform algebraic simplifications; for example, cancel out terms that appear in both the numerator and the denominator.

In the provided solution, this process involves simplifying:

• Start with \( f'(x) = \frac{12x^4 - 2x^2 - 8x^4 + 4x^2 - 2x}{x^4} \).
• Combine like terms in the numerator to simplify to: \( 4x^4 + 2x^2 - 2x \).
• Then, divide each term by \( x^4 \) to achieve the final simplified form \( f'(x) = 4 + \frac{2}{x^2} - \frac{2}{x^3} \).

The key is to achieve a simplified, clean expression that reflects the rate of change of the original function in a clear and interpretable way.