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Definition of the Derivative

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Definition of the Derivative

1. Core Concepts

The **Derivative** is the cornerstone of calculus, quantifying how fast a function changes at a specific point.

* **Physical Meaning**: Instantaneous Rate of Change (e.g., Velocity is the derivative of Displacement).

* **Geometric Meaning**: The **Slope of the Tangent Line** to the curve at a specific point.

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2. Definition and Formula

Let $y = f(x)$ be defined near $x_0$. When $x$ changes by $\Delta x$, $y$ changes by $\Delta y = f(x_0 + \Delta x) - f(x_0)$.

If the limit exists, the function is **Differentiable** at $x_0$, and the limit is called the **Derivative**:

$$ f'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} $$

**Notation**: $f'(x_0)$, $y'|_{x=x_0}$, or $\frac{dy}{dx}|_{x=x_0}$.

**Intuition**:

1. $\frac{\Delta y}{\Delta x}$ is the slope of the secant line, representing the **Average Rate of Change**.

2. As $\Delta x \to 0$, the secant line approaches the tangent line, and the average rate becomes the **Instantaneous Rate of Change**.

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3. Three-Step Method to Find the Derivative

1. **Find Increment**: $\Delta y = f(x_0+\Delta x) - f(x_0)$

2. **Find Ratio**: $\frac{\Delta y}{\Delta x}$ (Simplify the expression, usually cancelling $\Delta x$ in the denominator)

3. **Take Limit**: $\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$

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4. Typical Examples

**Example 1: Polynomial**

Find the derivative of $f(x) = x^2$ at $x=1$ using the definition.

**Solution**:

1. $\Delta y = (1+\Delta x)^2 - 1^2 = 2\Delta x + (\Delta x)^2$

2. $\frac{\Delta y}{\Delta x} = 2 + \Delta x$

3. $\lim_{\Delta x \to 0} (2 + \Delta x) = 2$

$\therefore f'(1) = 2$.

**Example 2: Rational Function**

Find the derivative of $f(x) = \frac{1}{x}$.

**Solution**:

1. $\Delta y = \frac{1}{x+\Delta x} - \frac{1}{x} = \frac{-\Delta x}{x(x+\Delta x)}$

2. $\frac{\Delta y}{\Delta x} = \frac{-1}{x(x+\Delta x)}$

3. $\lim_{\Delta x \to 0} \frac{-1}{x(x+\Delta x)} = -\frac{1}{x^2}$

$\therefore (\frac{1}{x})' = -\frac{1}{x^2}$.

**Example 3: Differentiability (Non-differentiable case)**

Discuss the differentiability of $f(x) = |x|$ at $x=0$.

**Solution**:

Check the limit $\lim_{\Delta x \to 0} \frac{|\Delta x|}{\Delta x}$.

* **Right Limit** ($\Delta x > 0$): $1$

* **Left Limit** ($\Delta x < 0$): $-1$

Since the limits are not equal, the derivative does not exist.

**Conclusion**: $f(x)=|x|$ is **not differentiable** at $x=0$ (Sharp corner).

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5. Common Pitfalls

* **Indeterminate Forms**: Always simplify the fraction $\frac{\Delta y}{\Delta x}$ before taking the limit to avoid $\frac{0}{0}$.

* **Continuity vs. Differentiability**: $y=|x|$ is continuous at $x=0$ but not differentiable. **Differentiability implies Continuity, but NOT vice versa**.

* **Notation**: Don't confuse the number $f'(x_0)$ with the function $f'(x)$.