2.3 Q-7

Question Statement

Differentiate the following expression with respect to $x$ :
$y = \frac{(1 + \sqrt{x})(x - x^{\frac{5}{2}})}{\sqrt{x}}$

This problem involves simplifying a rational expression before differentiating. The main steps include:

Simplifying the given fraction using basic algebraic identities.
Using differentiation rules for powers of $x$ $x$ :
- The derivative of $x^n$ is $n \cdot x^{n-1}$ .

The problem requires careful algebraic manipulation to make differentiation straightforward.

The given expression is:
$y = \frac{(1 + \sqrt{x})(x - x^{\frac{5}{2}})}{\sqrt{x}}$

Expand the numerator term by term: $y = \frac{(1 + \sqrt{x}) \cdot x(1 - \sqrt{x})}{\sqrt{x}}$

Using the difference of squares:
$(1 + \sqrt{x})(1 - \sqrt{x}) = 1 - x$

So: $y = \frac{x(1 - x)}{\sqrt{x}}$

Simplify further by splitting the terms: $y = \sqrt{x}(1 - x)$

Expand: $y = \sqrt{x} - x \sqrt{x}$

Convert to powers of $x$ : $y = x^{\frac{1}{2}} - x^{\frac{3}{2}}$

The expression is now: $y = x^{\frac{1}{2}} - x^{\frac{3}{2}}$

Differentiate term by term using the power rule:

The derivative of $x^{\frac{1}{2}}$ is:
$\frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}}$
The derivative of $x^{\frac{3}{2}}$ is:
$\frac{3}{2}x^{\frac{1}{2}}$

So: $\frac{dy}{dx} = \frac{1}{2\sqrt{x}} - \frac{3}{2}x^{\frac{1}{2}}$

Combine the terms:
$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1 - 3x}{\sqrt{x}}$

Final Answer:
$\frac{dy}{dx} = \frac{1 - 3x}{2\sqrt{x}}$

Simplification of Rational Expressions: $\frac{(1 + \sqrt{x})(1 - \sqrt{x})}{\sqrt{x}} \to 1 - x$
Power Rule for Differentiation: $\frac{d}{dx}(x^n) = n \cdot x^{n-1}$

Simplify the given expression:
$y = \sqrt{x} - x \sqrt{x} = x^{\frac{1}{2}} - x^{\frac{3}{2}}$
Differentiate each term using the power rule:
- $\frac{d}{dx}\left(x^{\frac{1}{2}}\right) = \frac{1}{2\sqrt{x}}$
- $\frac{d}{dx}\left(x^{\frac{3}{2}}\right) = \frac{3}{2}x^{\frac{1}{2}}$
Combine and simplify the result:
$\frac{dy}{dx} = \frac{1 - 3x}{2\sqrt{x}}$