Problem 15 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\). \(f(x)=\frac{(2 \sqrt{x}+1)(x-1)}{x+3}\)

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{2x^{3/2} + 8x^{1/2} - 6x^{-1/2} + 4}{(x+3)^2} \).

Step 1: Identify the Function to Differentiate

We are tasked with finding the derivative of the function \[ f(x) = \frac{(2 \sqrt{x} + 1)(x - 1)}{x + 3} \]This is a rational function where both the numerator and the denominator are polynomials or polynomial-like expressions.

Step 2: Use the Quotient Rule

To find the derivative of a rational function, we use the quotient rule. If we have \[ \frac{u}{v} \] the quotient rule states that \[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \] In this case, let \[ u = (2 \sqrt{x} + 1)(x - 1) \] and \[ v = x + 3 \].

Step 3: Differentiate the Numerator

First, calculate the derivative of the numerator. The numerator is \[ u = (2x^{1/2} + 1)(x - 1) \]Expand \[ u = 2x^{3/2} - 2x^{1/2} + x - 1 \]Differentiate \[ u' = \frac{d}{dx}(2x^{3/2}) + \frac{d}{dx}(-2x^{1/2}) + \frac{d}{dx}(x) + \frac{d}{dx}(-1) \]Therefore \[ u' = 3x^{1/2} - x^{-1/2} + 1 \].

Step 4: Differentiate the Denominator

Now differentiate the denominator \[ v = x + 3 \]Its derivative is straightforward: \[ v' = 1 \].

Step 5: Apply the Quotient Rule

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

Step 6: Simplify the Expression

Expand and simplify the numerator:\[ (3x^{1/2} - x^{-1/2} + 1)(x + 3) = 3x^{3/2} - x^{1/2} + x + 9x^{1/2} - 3x^{-1/2} + 3 \]Therefore, the full expression combined would require collecting like terms before placing it over \[ (x+3)^2 \].The final derivative of the function simplifies to \[ f'(x) = \frac{2x^{3/2} + 8x^{1/2} - 6x^{-1/2} + 4}{(x+3)^2} \].

Key Concepts

Quotient Rule

The Quotient Rule is a powerful tool in calculus for finding the derivative of a rational function, which is a fraction where both the numerator and the denominator are functions themselves. If you have a function expressed as \( \frac{u}{v} \), the rule provides a systematic way to find its derivative. The formula for the Quotient Rule is:

• \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)

Here, \( u \) is the numerator, \( v \) is the denominator, \( u' \) is the derivative of the numerator, and \( v' \) is the derivative of the denominator. This formula helps us compute derivatives cleanly by handling each part separately.When applying the Quotient Rule, it's essential to:

• Ensure you correctly compute both \( u' \) and \( v' \).
• Substitute these values back into the Quotient Rule formula accurately.
• Simplify your answer to get a more solvable expression.

This rule is particularly useful when dealing with rational functions, making differentiation straightforward and organized.

Differentiation Techniques

Differentiation is a core concept in calculus used to determine how a function changes at any given point. There are several techniques for differentiating functions, but when it comes to rational functions like the one given, specific strategies are helpful.One common technique is the use of the Quotient Rule, which we've discussed earlier. However, before you even apply this rule, it's often necessary to differentiate components of the function separately. For instance:

• Use Power Rule: To differentiate terms like \( 2x^{3/2} \), apply the power rule, which states if \( u = x^n \), then \( u' = nx^{n-1} \).
• Multiply through: When faced with expressions like \((2\sqrt{x} + 1)(x-1)\), first expand and simplify before differentiating.
• Constant Terms: Consider constant terms like \(-1\), where their derivative is simply 0.

To master differentiation, practice is key, as it helps you become nimble in recognizing which rules to apply and when to simplify terms for more straightforward calculation.

Simplifying Expressions

After applying differentiation techniques, it is crucial to simplify the resultant expression to make the derivative cleaner and easier to interpret. This process involves several key steps.For the given exercise, once you've applied the Quotient Rule, you will end up with a complex fraction in the numerator. The next immediate step is to:

• Expand brackets: Any multiplication of expanded terms should be done initially to reduce complexity.
• Combine like terms: Terms like \( 3x^{3/2} \) and \(- 2x^{3/2} \) should be combined to simplify the polynomial terms.
• Simplify fractions: Once the numerator terms are condensed, align over the common denominator expressed as a power, \( (x+3)^2 \), and see if any common factors can cancel out.

By simplifying your expressions, you make further calculations light work, enabling easier problem-solving and a better understanding of the function's behavior through its derivative.