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To address your request, I will solve the exercises step by step. Let’s begin with **Exercise 8B** and then move to **Exercise 8C**.

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### **Exercise 8B: Differentiation**
We are tasked with finding \( f'(x) \) for the given functions. I'll solve each one.

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#### **a.** \( f(x) = 12x + x^2 + 5 \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(12x) + \frac{d}{dx}(x^2) + \frac{d}{dx}(5)
\]
\[
f'(x) = 12 + 2x + 0 = 12 + 2x
\]

---

#### **b.** \( f(x) = 6x^5 + 6x - 7 \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(6x^5) + \frac{d}{dx}(6x) + \frac{d}{dx}(-7)
\]
\[
f'(x) = 30x^4 + 6 + 0 = 30x^4 + 6
\]

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#### **c.** \( f(x) = \frac{1}{2}x^{-1} - \frac{1}{2}x^{-2} \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}\left(\frac{1}{2}x^{-1}\right) + \frac{d}{dx}\left(-\frac{1}{2}x^{-2}\right)
\]
\[
f'(x) = -\frac{1}{2}x^{-2} + x^{-3} = -\frac{1}{2x^2} + \frac{1}{x^3}
\]

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#### **d.** \( f(x) = 4x^3 + 3x^{1/2} + 10x^{-1} \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(4x^3) + \frac{d}{dx}(3x^{1/2}) + \frac{d}{dx}(10x^{-1})
\]
\[
f'(x) = 12x^2 + \frac{3}{2}x^{-1/2} - 10x^{-2}
\]

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#### **e.** \( f(x) = 4 - 12x^{-2} + 4x^{1/2} \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(4) + \frac{d}{dx}(-12x^{-2}) + \frac{d}{dx}(4x^{1/2})
\]
\[
f'(x) = 0 + 24x^{-3} + 2x^{-1/2}
\]

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#### **f.** \( f(x) = 5x^{3/2} + 10x^{-2} \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(5x^{3/2}) + \frac{d}{dx}(10x^{-2})
\]
\[
f'(x) = \frac{15}{2}x^{1/2} - 20x^{-3}
\]

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#### **g.** \( f(x) = 9x^2 + x^2 + 4x + x^{-1} \)

Combine like terms first:
\[
f(x) = 10x^2 + 4x + x^{-1}
\]

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(10x^2) + \frac{d}{dx}(4x) + \frac{d}{dx}(x^{-1})
\]
\[
f'(x) = 20x + 4 - x^{-2}
\]

---

#### **h.** \( f(x) = 5x^{-2} + x^{-3} - 2x^{-1} \)

Differentiate term by term:
\[
f'(x) = \frac{d}{dx}(5x^{-2}) + \frac{d}{dx}(x^{-3}) + \frac{d}{dx}(-2x^{-1})
\]
\[
f'(x) = -10x^{-3} - 3x^{-4} + 2x^{-2}
\]

---

### **Exercise 8C: Integration**
Now, we will solve the integrals provided.

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#### **1.** \( \int (x^3 + 2x) \, dx \)

Integrate term by term:
\[
\int x^3 \, dx + \int 2x \, dx
\]
\[
\frac{x^4}{4} + x^2 + C
\]

---

#### **2.** \( \int (2x^{-2} + 3) \, dx \)

Integrate term by term:
\[
\int 2x^{-2} \, dx + \int 3 \, dx
\]
\[
-2x^{-1} + 3x + C
\]

---

#### **3.** \( \int (5x^{3/2} - 3x^2) \, dx \)

Integrate term by term:
\[
\int 5x^{3/2} \, dx - \int 3x^2 \, dx
\]
\[
\frac{10}{5}x^{5/2} - x^3 + C
\]

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#### **4.** \( \int (2x^{1/2} - 2x^{-1/2} + 4) \, dx \)

Integrate term by term:
\[
\int 2x^{1/2} \, dx - \int 2x^{-1/2} \, dx + \int 4 \, dx
\]
\[
\frac{4}{3}x^{3/2} - 4x^{1/2} + 4x + C
\]

---

#### **5.** \( \int (4x^3 - 3x^{-4} + r) \, dx \)

Integrate term by term (treat \( r \) as a constant):
\[
\int 4x^3 \, dx - \int 3x^{-4} \, dx + \int r \, dx
\]
\[
x^4 + \frac{3}{-3x^3} + rx + C = x^4 + r(x)Would you like me to continue solving the rest of **Exercise 8C** or clarify any of the above steps? Let me know! Below are some questions to deepen your understanding:

1. What is the difference between integration and differentiation?
2. How do you apply the power rule for derivatives and integrals?
3. Why do we add a constant \( C \) during indefinite integration?
4. Can you simplify fractional exponents, such as \( x^{3/2} \), before or after differentiation/integration?
5. What are some practical applications of integration in real-world scenarios?

**Tip**: Always rewrite complex terms with fractional or negative exponents before performing calculus operations. This simplifies the process and reduces errors.

This webpage provides detailed solutions to a series of differentiation and integration exercises. Learn how to apply the power rule for derivatives and integrals, and explore step-by-step solutions to various problems. Suitable for high school students in Grades 11-12.

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