2.3 Q-6

Question Statement

Differentiate the following expression with respect to $x$ :
$Y = \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2$

Background and Explanation

This problem involves the differentiation of a squared expression. To solve it, we will:

Expand the square using the formula:
$(a - b)^2 = a^2 + b^2 - 2ab$
Simplify the resulting expression.
Differentiate the simplified expression term by term using the basic differentiation rules:
- The derivative of $x$ is $1$ .
- The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$ .

Solution

Step 1: Expand the square

Given: $Y = \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2$

Expand using the formula $(a - b)^2 = a^2 + b^2 - 2ab$ :
$Y = (\sqrt{x})^2 + \left( \frac{1}{\sqrt{x}} \right)^2 - 2 \sqrt{x} \cdot \frac{1}{\sqrt{x}}$

Simplify each term:

$(\sqrt{x})^2 = x$
$\left( \frac{1}{\sqrt{x}} \right)^2 = \frac{1}{x}$
$2 \sqrt{x} \cdot \frac{1}{\sqrt{x}} = 2$

Substitute back: $Y = x + \frac{1}{x} - 2$

Step 2: Differentiate with respect to $x$

Now differentiate $Y$ term by term:

The derivative of $x$ is $1$ .
The derivative of $\frac{1}{x}$ is $-\frac{1}{x^2}$ .
The derivative of the constant $-2$ is $0$ .

So: $\frac{dY}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}\left( \frac{1}{x} \right) - \frac{d}{dx}(2)$ $\frac{dY}{dx} = 1 - \frac{1}{x^2} - 0$

Step 3: Simplify the result

Combine terms: $\frac{dY}{dx} = \frac{x^2 - 1}{x^2}$

Final Answer:
$\frac{dY}{dx} = \frac{x^2 - 1}{x^2}$

Key Formulas or Methods Used

Expansion of Squares:
$(a - b)^2 = a^2 + b^2 - 2ab$
Basic Derivatives:
- $\frac{d}{dx}(x) = 1$
- $\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}$
- The derivative of a constant is $0$ .

Summary of Steps

Expand the square:
$\left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 \to x + \frac{1}{x} - 2$
Differentiate each term:
- $\frac{d}{dx}(x) = 1$
- $\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}$
- $\frac{d}{dx}(-2) = 0$
Combine the results:
$\frac{dY}{dx} = \frac{x^2 - 1}{x^2}$