Problem 11 - Calculus - AP Edition

Question

$$\text { Find } f^{\prime}(x)$$. $$f(x)=\frac{3 x+4}{x^{2}+1}$$

Step-by-Step Solution

Answer

\( f'(x) = \frac{-3x^2 - 8x + 3}{(x^2 + 1)^2} \).

Step 1: Identify the Function Form

The function given is \( f(x) = \frac{3x + 4}{x^2 + 1} \), which is in the form of \( \frac{u}{v} \), where \( u = 3x + 4 \) and \( v = x^2 + 1 \). This requires the use of the quotient rule to find the derivative.

Step 2: Recall the Quotient Rule

The quotient rule states that if you have a function \( y = \frac{u}{v} \), then the derivative \( y' \) is given by \( y' = \frac{u'v - uv'}{v^2} \). We will apply this formula by first finding \( u' \) and \( v' \).

Step 3: Differentiate the Numerator

Find the derivative of the numerator \( u = 3x + 4 \). Using basic differentiation rules, \( u' = \frac{d}{dx}(3x + 4) = 3 \).

Step 4: Differentiate the Denominator

Find the derivative of the denominator \( v = x^2 + 1 \). Using basic differentiation rules, \( v' = \frac{d}{dx}(x^2 + 1) = 2x \).

Step 5: Apply the Quotient Rule

Using the quotient rule \( f'(x) = \frac{u'v - uv'}{v^2} \), substitute \( u' = 3 \), \( v = x^2 + 1 \), \( u = 3x + 4 \), and \( v' = 2x \) to get:\[ f'(x) = \frac{3(x^2 + 1) - (3x + 4)(2x)}{(x^2 + 1)^2} \]

Step 6: Simplify the Expression

Simplify the expression in the numerator:\[ 3(x^2 + 1) - (3x + 4)(2x) = 3x^2 + 3 - (6x^2 + 8x) \]\[ = 3x^2 + 3 - 6x^2 - 8x \]Combine like terms:\[ = -3x^2 - 8x + 3 \]So, \[ f'(x) = \frac{-3x^2 - 8x + 3}{(x^2 + 1)^2} \].

Key Concepts

Derivative of Rational Functions

The given function, \( f(x) = \frac{3x + 4}{x^2 + 1} \), is a rational function. Rational functions are ratios of two polynomials, and their derivatives can often be challenging. Fortunately, techniques like the Quotient Rule help us simplify this process.

Unlike simpler polynomial or exponential functions, rational functions require more care during differentiation due to their setup as fractions. The derivative of a rational function provides insights into its rate of change and behavior across its domain, particularly in determining features such as maxima, minima, and points of inflection.

In this specific problem, the function is arranged such that, by identifying the numerator and the denominator along with their derivatives, we are armed with the necessary components for applying differentiation rules effectively.

Differentiation Techniques

To differentiate rational functions like \( f(x) = \frac{3x + 4}{x^2 + 1} \), we employ the Quotient Rule—an essential technique in calculus.

The Quotient Rule states that if a function \( y = \frac{u}{v} \), its derivative is \[ y' = \frac{u'v - uv'}{v^2} \]Where:

• \( u \) is the numerator (e.g., \( 3x+4 \))
• \( v \) is the denominator (e.g., \( x^2+1 \))
• \( u' \) and \( v' \) are their respective derivatives

Understanding and correctly applying such differentiation techniques is vital, as they permit us to navigate more complex expressions. For polynomials, basic derivative rules apply, making it easier to calculate \( u' \) and \( v' \). Here, the numerator differentiates to 3, while the denominator differentiates to \( 2x \). This illustrates the importance of breaking down each part of the function to methodically apply the overarching rule.

Each small differentiation step builds the pathway to solving a broader calculus problem, highlighting the significance of these foundational techniques.

Calculus Problem Solving

Calculus problem-solving involves identifying the components needed to address a given problem and applying systematic rules as seen in this exercise.

After computing derivatives, the next step is substitution into the Quotient Rule:\[ f'(x) = \frac{3(x^2 + 1) - (3x + 4)(2x)}{(x^2 + 1)^2} \]Breaking down the expression and simplifying are crucial to finding the correct derivative, helping verify the accuracy of each calculation step.

In paper examination settings or real-world application, identifying potential errors or simplification mishaps can save significant hassle later on. Such simplification might involve expanding expressions and being careful with terms, as seen when reducing \( 3x^2 + 3 - 6x^2 - 8x \) to \( -3x^2 - 8x + 3 \).

Using organized steps and clear logic builds confidence and enhances competence in solving derivative problems, reinforcing understanding of calculus principles and helping tackle increasingly challenging mathematical problems.