2.3 Q-16

Question Statement

Given that $y = \sqrt{x} - \frac{1}{\sqrt{x}}$, prove that:

$$2x \frac{dy}{dx} + y = 2\sqrt{x}$$

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Background and Explanation

This problem involves differentiation of a function in terms of $x$ and requires us to apply the rules of differentiation to both terms in the expression for $y$. We will use basic differentiation techniques, such as the power rule, to differentiate $y$ with respect to $x$, then substitute and simplify to reach the desired result.

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Solution

Let’s break down the steps:

1. Start with the given expression for $y$:

$$y = \sqrt{x} - \frac{1}{\sqrt{x}}$$

2. Differentiate $y$ with respect to $x$:

   We will apply the power rule to differentiate each term in the expression.

$$\frac{dy}{dx} = \frac{d}{dx} \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)$$

The derivative of $\sqrt{x}$ is $\frac{1}{2} x^{-\frac{1}{2}}$, and the derivative of $\frac{1}{\sqrt{x}}$ is $\frac{1}{2} x^{-\frac{3}{2}}$ with a negative sign.

Therefore, we get:

$$\frac{dy}{dx} = \frac{1}{2} x^{-\frac{1}{2}} + \frac{1}{2} x^{-\frac{3}{2}}$$

3. Multiply both sides of the derivative by 2:

$$2 \frac{dy}{dx} = \frac{1}{\sqrt{x}} + \frac{1}{x \sqrt{x}}$$

4. Multiply the result by $x$ to compute $2x \frac{dy}{dx}$:

$$2x \frac{dy}{dx} = x \cdot \frac{1}{\sqrt{x}} + x \cdot \frac{1}{x \sqrt{x}} = \sqrt{x} + \frac{1}{\sqrt{x}}$$

5. Substitute the value of $y$ back into the equation:

   Since $y = \sqrt{x} - \frac{1}{\sqrt{x}}$, we can substitute this into the expression:

$$2x \frac{dy}{dx} + y = \sqrt{x} + \frac{1}{\sqrt{x}} + \sqrt{x} - \frac{1}{\sqrt{x}}$$

6. Simplify the terms:

   The terms $\frac{1}{\sqrt{x}}$ and $-\frac{1}{\sqrt{x}}$ cancel each other out, leaving us with:

$$2x \frac{dy}{dx} + y = 2\sqrt{x}$$

Hence, we have proved the given statement.

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Key Formulas or Methods Used

• Power Rule of Differentiation:

$$\frac{d}{dx} x^n = n x^{n-1}$$

• Substitution of the given value of $y$ to complete the proof.

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Summary of Steps

1. Write the expression for $y$.
2. Differentiate $y$ with respect to $x$.
3. Multiply the derivative by 2 to get $2 \frac{dy}{dx}$.
4. Multiply by $x$ to obtain $2x \frac{dy}{dx}$.
5. Substitute the value of $y$ into the equation.
6. Simplify and prove that $2x \frac{dy}{dx} + y = 2\sqrt{x}$.