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Let's solve these differentiation problems one by one:

**(a) \( y = x^2 \)**
\[
\frac{dy}{dx} = 2x
\]

**(b) \( y = x^5 \)**
\[
\frac{dy}{dx} = 5x^4
\]

**(c) \( y = x^{-1} \)**
\[
\frac{dy}{dx} = -x^{-2} = -\frac{1}{x^2}
\]

**(d) \( y = x^{1/2} \)**
\[
\frac{dy}{dx} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}
\]

**(e) \( y = \pi \)**
\[
\frac{dy}{dx} = 0 \quad \text{(since \(\pi\) is a constant)}
\]

**(f) \( y = 3x^4 \)**
\[
\frac{dy}{dx} = 3 \cdot 4x^3 = 12x^3
\]

**(g) \( y = x^2 + 3 \)**
\[
\frac{dy}{dx} = 2x + 0 = 2x
\]

**(h) \( y = x^5 - 6x^7 \)**
\[
\frac{dy}{dx} = 5x^4 - 6 \cdot 7x^6 = 5x^4 - 42x^6
\]

**(i) \( y = \frac{2x^2 - 3}{\sqrt{x}} \)**

First, rewrite \( y \) as:
\[
y = \frac{2x^2 - 3}{x^{1/2}} = 2x^{3/2} - 3x^{-1/2}
\]
Now, differentiate:
\[
\frac{dy}{dx} = 2 \cdot \frac{3}{2}x^{1/2} - 3 \cdot \left(-\frac{1}{2}\right)x^{-3/2}
\]
\[
\frac{dy}{dx} = 3x^{1/2} + \frac{3}{2}x^{-3/2}
\]

**(j) \( y = (x + 1)(x^2 + 3) \)**

Use the product rule:
\[
\frac{dy}{dx} = (x+1) \frac{d}{dx}(x^2+3) + (x^2+3) \frac{d}{dx}(x+1)
\]
\[
= (x+1)(2x) + (x^2+3)(1)
\]
\[
= 2x(x+1) + (x^2 + 3) = 2x^2 + 2x + x^2 + 3
\]
\[
\frac{dy}{dx} = 3x^2 + 2x + 3
\]

**(k) \( y = \frac{3x^2 - 1}{x} \)**

First, rewrite \( y \) as:
\[
y = \frac{3x^2}{x} - \frac{1}{x} = 3x - x^{-1}
\]
Now, differentiate:
\[
\frac{dy}{dx} = 3 + x^{-2} = 3 + \frac{1}{x^2}
\]

**(l) \( y = \sqrt{x(x+1)} \)**

First, rewrite \( y \) as:
\[
y = (x(x+1))^{1/2} = (x^2 + x)^{1/2}
\]
Now, apply the chain rule:
\[
\frac{dy}{dx} = \frac{1}{2}(x^2 + x)^{-1/2} \cdot (2x + 1)
\]
\[
\frac{dy}{dx} = \frac{2x+1}{2\sqrt{x^2 + x}}
\]

Would you like more details on any of these steps or have any questions?

Here are some related problems to consider:
1. Differentiate \( y = x^3 \).
2. Differentiate \( y = \sin(x) \).
3. Differentiate \( y = e^x \).
4. Differentiate \( y = \ln(x) \).
5. Differentiate \( y = \frac{1}{x+2} \).

**Tip:** Remember to always simplify functions before differentiating, if possible!

Differentiate the following functions: (a) y = x^2, (b) y = x^5, (c) y = x^{-1}, (d) y = x^{1/2}, (e) y = π, (f) y = 3x^4, (g) y = x^2 + 3, (h) y = x^5 - 6x^7, (i) y = (2x^2 - 3)/√x, (j) y = (x + 1)(x^2 + 3), (k) y = (3x^2 - 1)/x, (l) y = √x(x + 1).

This set of differentiation problems covers key calculus concepts, including the power rule, chain rule, and product rule. Learn to differentiate functions like y = x^5, y = (x + 1)(x^2 + 3), and more with step-by-step solutions. Suitable for college-level calculus students.

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