Differentiate (2x−3)(x⎯⎯√+1)divided  by x⎯⎯√ with respect to x.

To differentiate \(\frac{(2x - 3)(\sqrt{x} + 1)}{\sqrt{x}}\) with respect to \(x\), we will first simplify the expression and then apply the rules of differentiation.

Given:
\[
\frac{(2x - 3)(\sqrt{x} + 1)}{\sqrt{x}}
\]

1. Simplify the expression inside the fraction:
\[
(2x - 3)(\sqrt{x} + 1) = (2x - 3)\sqrt{x} + (2x - 3)
\]

2. Distribute \((2x - 3)\):
\[
= 2x\sqrt{x} - 3\sqrt{x} + 2x - 3
\]

3. Divide each term by \(\sqrt{x}\):
\[
\frac{2x\sqrt{x}}{\sqrt{x}} - \frac{3\sqrt{x}}{\sqrt{x}} + \frac{2x}{\sqrt{x}} - \frac{3}{\sqrt{x}}
\]

4. Simplify each term:
\[
= 2x - 3 + 2\sqrt{x} - \frac{3}{\sqrt{x}}
\]

Now, we have a simplified expression to differentiate:
\[
2x - 3 + 2\sqrt{x} - \frac{3}{\sqrt{x}}
\]

Next, we differentiate each term separately with respect to \(x\):

\[
\frac{d}{dx}(2x) = 2
\]
\[
\frac{d}{dx}(-3) = 0
\]
\[
\frac{d}{dx}(2\sqrt{x}) = \frac{d}{dx}(2x^{1/2}) = 2 \cdot \frac{1}{2}x^{-1/2} = \frac{1}{\sqrt{x}}
\]
\[
\frac{d}{dx}\left(-\frac{3}{\sqrt{x}}\right) = \frac{d}{dx}\left(-3x^{-1/2}\right) = -3 \cdot \left(-\frac{1}{2}x^{-3/2}\right) = \frac{3}{2}x^{-3/2} = \frac{3}{2x^{3/2}}
\]

Combine the differentiated terms:
\[
2 + 0 + \frac{1}{\sqrt{x}} + \frac{3}{2x^{3/2}}
\]

Thus, the derivative of \(\frac{(2x - 3)(\sqrt{x} + 1)}{\sqrt{x}}\) with respect to \(x\) is:
\[
2 + \frac{1}{\sqrt{x}} + \frac{3}{2x^{3/2}}
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions you might want to ask:
1. How do you differentiate \((2x - 3)(\sqrt{x} + 1)\) without simplifying first?
2. What are some common rules of differentiation?
3. How do you simplify \(\frac{d}{dx}\left(\frac{3}{\sqrt{x}}\right)\)?
4. What is the chain rule in differentiation?
5. How do you handle the differentiation of products of functions?

**Tip:** Always simplify the expression as much as possible before differentiating to make the process easier.

Learn how to differentiate the expression (2x−3)(√x+1)/√x with respect to x. This problem involves applying the product rule and simplifying complex expressions for differentiation. Suitable for advanced high school students and college-level calculus courses.

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