### Differentiation: An Overview

**Differentiation** is a fundamental concept in calculus. It deals with finding the rate at which a function is changing at any given point. In other words, it measures how a function's output value changes as its input value changes. The process of finding the derivative of a function is called **differentiation**.

The derivative of a function \( f(x) \) at a point gives the **slope of the tangent line** to the curve of the function at that point.

### Basic Rules of Differentiation

1. **Power Rule**:  
   If \( f(x) = x^n \), where \( n \) is a constant, then:
   \[
   \frac{d}{dx}[x^n] = nx^{n-1}
   \]
   Example:  
   If \( f(x) = x^3 \), then:
   \[
   \frac{d}{dx}[x^3] = 3x^2
   \]

2. **Constant Rule**:  
   The derivative of a constant is zero. If \( f(x) = c \), where \( c \) is a constant, then:
   \[
   \frac{d}{dx}[c] = 0
   \]
   Example:  
   If \( f(x) = 5 \), then:
   \[
   \frac{d}{dx}[5] = 0
   \]

3. **Sum Rule**:  
   If \( f(x) = g(x) + h(x) \), then:
   \[
   \frac{d}{dx}[f(x)] = \frac{d}{dx}[g(x)] + \frac{d}{dx}[h(x)]
   \]
   Example:  
   If \( f(x) = 2x^2 + 3x \), then:
   \[
   \frac{d}{dx}[f(x)] = \frac{d}{dx}[2x^2] + \frac{d}{dx}[3x] = 4x + 3
   \]

4. **Product Rule**:  
   If \( f(x) = g(x) \cdot h(x) \), then:
   \[
   \frac{d}{dx}[f(x)] = g'(x) \cdot h(x) + g(x) \cdot h'(x)
   \]
   Example:  
   If \( f(x) = x^2 \cdot \sin(x) \), then:
   \[
   \frac{d}{dx}[x^2 \cdot \sin(x)] = 2x \cdot \sin(x) + x^2 \cdot \cos(x)
   \]

5. **Quotient Rule**:  
   If \( f(x) = \frac{g(x)}{h(x)} \), then:
   \[
   \frac{d}{dx}[f(x)] = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2}
   \]
   Example:  
   If \( f(x) = \frac{x^2}{x+1} \), then:
   \[
   \frac{d}{dx} \left[ \frac{x^2}{x+1} \right] = \frac{(2x) \cdot (x+1) - (x^2) \cdot (1)}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}
   \]

6. **Chain Rule**:  
   If \( f(x) = g(h(x)) \), then:
   \[
   \frac{d}{dx}[f(x)] = g'(h(x)) \cdot h'(x)
   \]
   Example:  
   If \( f(x) = \sin(x^2) \), then:
   \[
   \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x
   \]

---

### Examples of Differentiation

#### Example 1:  
Find the derivative of \( f(x) = 5x^4 - 3x^3 + 2x - 7 \).

Solution:
Using the power rule:
\[
\frac{d}{dx}[f(x)] = 20x^3 - 9x^2 + 2
\]

#### Example 2:
Find the derivative of \( f(x) = \frac{3x^2 + 2x}{x - 1} \).

Solution:
Use the quotient rule:
\[
\frac{d}{dx} \left[ \frac{3x^2 + 2x}{x - 1} \right] = \frac{(6x + 2)(x - 1) - (3x^2 + 2x)(1)}{(x - 1)^2}
\]
Simplify the result.

#### Example 3:
Find the derivative of \( f(x) = \tan(3x) \).

Solution:
Using the chain rule:
\[
\frac{d}{dx}[\tan(3x)] = \sec^2(3x) \cdot 3 = 3 \sec^2(3x)
\]

---

### Want More Details or Have Questions?

If you’d like more specific examples or want further clarification on any rules, feel free to ask.

#### Further Questions:
1. How does the product rule differ from the quotient rule in terms of application?
2. What does the derivative of a function tell us about its graph?
3. Can we differentiate trigonometric functions in the same way as polynomial functions?
4. What happens if we apply the power rule to functions involving negative exponents?
5. How do implicit differentiation techniques work when dealing with equations that don’t have an explicit \( y = f(x) \) form?

**Tip:**  
To improve your understanding of differentiation, always practice applying the rules to a variety of functions (polynomials, trigonometric, exponential). The more you practice, the easier it becomes to identify which rule to apply!

Learn the fundamentals of differentiation, including key rules such as the power rule, product rule, quotient rule, and chain rule. Explore step-by-step examples of differentiation applied to polynomial, trigonometric, and rational functions. Suitable for students in Grades 11-12 and college-level calculus courses.

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