Derivatives

Derivatives - Essential Concepts

The derivative represents a function's rate of change. Here's what you need to know, condensed but comprehensive:

Definition of the Derivative

The derivative of $$f(x)$$ is defined as:

\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Geometric interpretation: The derivative at a point is the slope of the tangent line at that point.

Basic Differentiation Rules - Your Essential Toolkit

1. Constant Rule: If $$f(x) = c$$, then $$f'(x) = 0$$
2. Power Rule: If $$f(x) = x^n$$, then $$f'(x) = n \cdot x^{n-1}$$
3. Sum Rule: If $$f(x) = g(x) + h(x)$$, then $$f'(x) = g'(x) + h'(x)$$
4. Product Rule: If $$f(x) = g(x) \cdot h(x)$$, then $$f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)$$
5. Quotient Rule: If $$f(x) = \frac{g(x)}{h(x)}$$, then $$f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{[h(x)]^2}$$
6. Chain Rule: If $$f(x) = g(h(x))$$, then $$f'(x) = g'(h(x)) \cdot h'(x)$$

Common derivatives to memorize:

• $$\frac{d}{dx}(\sin x) = \cos x$$
• $$\frac{d}{dx}(\cos x) = -\sin x$$
• $$\frac{d}{dx}(e^x) = e^x$$
• $$\frac{d}{dx}(\ln x) = \frac{1}{x}$$

Examples - Applying the Rules

Example 1: Find the derivative of $$f(x) = 3x^2 + 2x - 5$$

Solution:

\[f'(x) = 3 \cdot 2x^{2-1} + 2 \cdot 1x^{1-1} - 0 = 6x + 2\]

Example 2: Find the derivative of $$f(x) = x^3 \cdot (2x + 1)$$

Solution using the Product Rule:

\[f'(x) = (3x^2) \cdot (2x + 1) + x^3 \cdot 2 = 6x^3 + 3x^2 + 2x^3 = 8x^3 + 3x^2\]

Key insight: When differentiating complex functions, break them down using these rules. Most calculus problems can be solved by correctly applying these fundamental rules.