Problem 24 - University Calculus: Early Transcendentals

Question

Find the derivatives of the functions. $$u=\frac{5 x+1}{2 \sqrt{x}}$$

Step-by-Step Solution

Answer

\( u' = \frac{5}{4}x^{-1/2} - \frac{1}{4}x^{-3/2} \)

Step 1: Recognize the Structure of the Function

The function given is \( u = \frac{5x + 1}{2\sqrt{x}} \). This is a rational function where the numerator is a linear expression and the denominator is a square root. We need to simplify this function first to make differentiation easier.

Step 2: Simplify the Function

Rewrite the square root in the denominator as an exponent: \( u = \frac{5x + 1}{2x^{1/2}} \). We can split this into separate terms: \( u = \frac{5x}{2x^{1/2}} + \frac{1}{2x^{1/2}} \). This simplifies to \( u = \frac{5}{2}x^{1/2} + \frac{1}{2}x^{-1/2} \).

Step 3: Apply Differentiation Rules

Differentiate each term using the power rule. For the first term, \( \frac{5}{2}x^{1/2} \), the derivative is \( \frac{5}{2} \cdot \frac{1}{2}x^{-1/2} = \frac{5}{4}x^{-1/2} \). For the second term, \( \frac{1}{2}x^{-1/2} \), the derivative is \( \frac{1}{2} \cdot \left(-\frac{1}{2}\right)x^{-3/2} = -\frac{1}{4}x^{-3/2} \).

Step 4: Combine the Derivatives

Combine the derivatives from Step 3 to get the final result: \( u' = \frac{5}{4}x^{-1/2} - \frac{1}{4}x^{-3/2} \).

Key Concepts

Rational Functions

Rational functions are mathematical expressions representing the division of two polynomials. In the given exercise, the function is \( u = \frac{5x + 1}{2\sqrt{x}} \). Here, the numerator is a linear polynomial \(5x + 1\) and the denominator is \(2\sqrt{x}\), which can be rewritten as \(2x^{1/2}\). When dealing with rational functions, the goal is to simplify the expression if possible, especially when preparing for differentiation or integration.

The simplification process often involves factoring, expanding, or rewriting components using exponent rules. This transforms the function into a form that is easily manageable for further calculations, like determining derivatives. By converting \(\sqrt{x}\) to \(x^{1/2}\), we can use the power rule more effectively later on. Rational functions also frequently arise in real-world applications where relationships between quantities are expressed as ratios.

Power Rule

The power rule is a fundamental tool in calculus used to find derivatives of polynomial expressions. This rule states that if you have a function in the form of \(x^n\), its derivative is \(nx^{n-1}\). For the function \(u = \frac{5}{2}x^{1/2} + \frac{1}{2}x^{-1/2}\), we apply the power rule to differentiate each term separately.

For the term \(\frac{5}{2}x^{1/2}\), using the power rule, we multiply the exponent \(\frac{1}{2}\) by the coefficient \(\frac{5}{2}\), and decrease the exponent by one, resulting in \(\frac{5}{4}x^{-1/2}\).

• Calculate \(\frac{5}{2} \times \frac{1}{2} = \frac{5}{4}\)
• Reduce exponent: \(\frac{1}{2} - 1 = -\frac{1}{2}\)

Similarly, for \(\frac{1}{2}x^{-1/2}\), the derivative is found by multiplying the coefficient \(\frac{1}{2}\) by \(-\frac{1}{2}\), and decreasing the exponent by one, resulting in \(-\frac{1}{4}x^{-3/2}\).

• Calculate \(\frac{1}{2} \times -\frac{1}{2} = -\frac{1}{4}\)
• Reduce exponent: \(-\frac{1}{2} - 1 = -\frac{3}{2}\)

The power rule simplifies derivative calculations for expressions raised to a power, allowing for efficient simplification and problem-solving.

Simplification of Expressions

Simplifying expressions before performing operations like differentiation makes the process much easier. In the exercise, the function \( u = \frac{5x + 1}{2\sqrt{x}} \) was initially complicated due to the division by a square root. Simplifying it by expressing terms of the form \(x^{n}\) is often a beneficial first step.

The given problem was simplified as follows: rewrite the denominator \(\sqrt{x}\) as \(x^{1/2}\). Then, separate the expression: \(\frac{5x}{2x^{1/2}} + \frac{1}{2x^{1/2}}\), which further simplifies using algebraic rules to \(\frac{5}{2}x^{1/2} + \frac{1}{2}x^{-1/2}\). This transformation stripped down the complexity, preparing it for differentiation.

• Convert radicals to exponents: \(\sqrt{x} = x^{1/2}\)
• Rewrite fractions for individual terms
• Simplify evenly via division: \(\frac{5x}{2x^{1/2}} = \frac{5}{2}x^{1/2}\)
• Express other terms in a similar format: \(\frac{1}{2x^{1/2}} = \frac{1}{2}x^{-1/2}\)

Understanding simplification techniques is crucial because it leads to more straightforward calculations and aids in revealing the structure of expressions.

Derivative Calculations

Calculating derivatives is a core part of calculus, focusing on understanding rates of change. The simplified function \( u = \frac{5}{2}x^{1/2} + \frac{1}{2}x^{-1/2} \) allows us to easily apply derivative rules.

Using the power rule, the derivative of \(\frac{5}{2}x^{1/2}\) was found to be \(\frac{5}{4}x^{-1/2}\). Similarly, \(\frac{1}{2}x^{-1/2}\) resulted in a derivative of \(-\frac{1}{4}x^{-3/2}\).

Combining these results provides the final derivative of the original function:

• \(\frac{d}{dx}\left(\frac{5}{2}x^{1/2}\right) = \frac{5}{4}x^{-1/2}\)
• \(\frac{d}{dx}\left(\frac{1}{2}x^{-1/2}\right) = -\frac{1}{4}x^{-3/2}\)
• Combine them to get: \(u' = \frac{5}{4}x^{-1/2} - \frac{1}{4}x^{-3/2}\)

Derivative calculations reveal how a function behaves as its input changes, crucial in numerous fields such as physics for motion analysis and biology for population growth patterns. Mastering these calculations involves practice and a solid grasp of fundamental rules.