Problem 22 - Thomas Calculus

Question

In Exercises \(19-30,\) find \(d y\) $$ y=\frac{2 \sqrt{x}}{3(1+\sqrt{x})} $$

Step-by-Step Solution

Answer

The differential is \( dy = \frac{3 - 3\sqrt{x}}{\sqrt{x}(9 + 18\sqrt{x} + 9x)} dx \).

Step 1: Identify the Function

The given function is \( y = \frac{2 \sqrt{x}}{3(1 + \sqrt{x})} \). We need to find the differential \( dy \).

Step 2: Differentiate Using the Quotient Rule

The Quotient Rule states that if \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \). In this problem, set \( u = 2\sqrt{x} \) and \( v = 3(1+\sqrt{x}) \). We will differentiate \( u \) and \( v \) individually.

Step 3: Differentiate \( u = 2\sqrt{x} \)

Differentiating \( u \) with respect to \( x \), we get \( \frac{du}{dx} = \frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{-1/2} = x^{-1/2} = \frac{1}{\sqrt{x}} \).

Step 4: Differentiate \( v = 3(1+\sqrt{x}) \)

Differentiating \( v \) with respect to \( x \), we get \( \frac{dv}{dx} = 3 \cdot \frac{d}{dx}(1+x^{1/2}) = 3 \cdot 0 + 3 \cdot \frac{1}{2}x^{-1/2} = \frac{3}{2}x^{-1/2} = \frac{3}{2\sqrt{x}} \).

Step 5: Apply the Quotient Rule

Apply the Quotient Rule: \( \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \). Substitute \( u = 2\sqrt{x} \), \( v = 3(1+\sqrt{x}) \), \( \frac{du}{dx} = \frac{1}{\sqrt{x}} \), and \( \frac{dv}{dx} = \frac{3}{2\sqrt{x}} \) into the formula.

Step 6: Substitute and Simplify

Substitute the derivatives into the Quotient Rule formula: \( \frac{dy}{dx} = \frac{3(1+\sqrt{x}) \cdot \frac{1}{\sqrt{x}} - 2\sqrt{x} \cdot \frac{3}{2\sqrt{x}}}{[3(1 + \sqrt{x})]^2} \). Simplify the numerator and the denominator separately.

Step 7: Compute the Numerator

The numerator becomes: \( 3 \cdot \frac{1+\sqrt{x}}{\sqrt{x}} - 3 = \frac{3 + 3\sqrt{x} - 3\sqrt{x}}{\sqrt{x}} = \frac{3}{\sqrt{x}} - 3 = \frac{3 - 3\sqrt{x}}{\sqrt{x}} \).

Step 8: Compute the Denominator and Simplify

The denominator is \( [3(1+\sqrt{x})]^2 = 9(1+2\sqrt{x}+x) = 9 + 18\sqrt{x} + 9x \). Thus, \( \frac{dy}{dx} = \frac{3(1 + \sqrt{x}) - 3\sqrt{x}}{x^{1/2}[3 + 18x^{1/2} + 9x]} = \frac{3 - 3\sqrt{x}}{x^{1/2}(9 + 18\sqrt{x} + 9x)} \).

Step 9: Express the Result

Express \( dy \) using the derivative: \( dy = \frac{3 - 3\sqrt{x}}{\sqrt{x}(9 + 18\sqrt{x} + 9x)} dx \). This is the differential of the function.

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is an essential tool for finding derivatives when dealing with functions that are ratios of two other functions. It allows us to differentiate a function that is written as a fraction, which occurs frequently in mathematical problems. For example, if you have a function in the form \( y = \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), the Quotient Rule enables you to find the derivative \( \frac{dy}{dx} \) efficiently.

The formula for the Quotient Rule is given by: \[ \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \] Where:

• \( v \) and \( u \) are the two functions that make up the fraction.
• \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) are their respective derivatives.
• \( v^2 \) is the denominator squared.

This rule helps by keeping the process systematic, and it’s crucial for ensuring accuracy, especially as the expressions become complex.

Differentiation

Differentiation is the process of finding the derivative of a function. The derivative measures how a function changes as its input changes. This is tremendously important in calculus and helps in understanding the dynamics of functions in terms of limits, rates of change, and tangent lines.

Differentiation allows you to find the slope of a function at any given point, which is crucial for applications like optimizations, finding maxima and minima, and solving physics problems. For the function \( y = 2\sqrt{x} \), differentiation enables us to find how it changes concerning \( x \). By using basic differentiation rules, such as the power rule, we can easily compute derivatives of different functions.

The basic formula for differentiation using the power rule, where \( n \) is any real number, is: \[ \frac{d}{dx}(x^n) = n \cdot x^{n-1} \] In the process outlined, we differentiated simple terms like \( 2x^{1/2} \) and \( 1 + x^{1/2} \). Understanding how these individual components change when \( x \) changes sets a strong foundation for deeper calculus problems.

Derivative Computation

Computing derivatives is about applying differentiation techniques effectively. In our exercise, we take the derivative of two specific parts of the function to apply the Quotient Rule accurately.

First, we compute the derivative of \( u = 2\sqrt{x} \). Using the power rule explained earlier, the derivative \( \frac{du}{dx} \) is found to be \( \frac{1}{\sqrt{x}} \). This involves rewriting the square root as a power \( (x^{1/2}) \) and applying the power rule.

Next, for \( v = 3(1 + \sqrt{x}) \), we differentiate to get \( \frac{dv}{dx} = \frac{3}{2\sqrt{x}} \). Here too, the power rule is used alongside the constant multiple rule, where the constant factor (in this case, the coefficient \( 3 \)) is factored out during differentiation.

Once the derivatives of \( u \) and \( v \) are calculated, the Quotient Rule synthesizes these parts into a single derivative expression. This derivative tells us how the entire fraction \( y = \frac{2 \sqrt{x}}{3(1 + \sqrt{x})} \) changes with respect to \( x \), providing insight into the function's behavior.