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Derivatives

Derivatives

The **derivative** of a function measures the **instantaneous rate of change** of the function with respect to its variable. It represents the **slope of the tangent line** at any given point.

1. Definition of a Derivative

The derivative of \( f(x) \), denoted as \( f'(x) \) or \( \frac{d}{dx} f(x) \), is defined as:

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

This limit must exist for \( f(x) \) to be **differentiable** at \( x \).

2. Basic Differentiation Rules

The following rules simplify differentiation:

  • Power Rule: \( \frac{d}{dx} x^n = n x^{n-1} \)
  • Constant Rule: \( \frac{d}{dx} c = 0 \) for any constant \( c \)
  • Constant Multiple Rule: \( \frac{d}{dx} [c f(x)] = c f'(x) \)
  • Sum Rule: \( \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) \)
  • Product Rule: \( \frac{d}{dx} [f(x) g(x)] = f(x) g'(x) + f'(x) g(x) \)
  • Quotient Rule: \( \frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2} \)
  • Chain Rule: \( \frac{d}{dx} f(g(x)) = f'(g(x)) g'(x) \)

3. Example Applications

Example 1: Using the Power Rule

Differentiate:

\[ f(x) = 5x^4 - 3x^2 + 7 \]

Applying the power rule:

\[ f'(x) = (5 \times 4)x^{3} - (3 \times 2)x^{1} + 0 \] \[ f'(x) = 20x^3 - 6x \]

Example 2: Product Rule

Find \( \frac{d}{dx} [(x^2 + 1)(x - 3)] \).

Using the product rule:

\[ \frac{d}{dx} [(x^2 + 1)(x - 3)] = (x^2 + 1)'(x - 3) + (x^2 + 1)(x - 3)' \] \[ = (2x)(x - 3) + (x^2 + 1)(1) \] \[ = 2x(x - 3) + x^2 + 1 \] \[ = 2x^2 - 6x + x^2 + 1 \] \[ = 3x^2 - 6x + 1 \]

Example 3: Chain Rule

Find \( \frac{d}{dx} \) of \( f(x) = (3x^2 + 5x)^4 \).

Using the chain rule:

\[ f'(x) = 4(3x^2 + 5x)^3 \times (6x + 5) \] \[ = 4(6x + 5)(3x^2 + 5x)^3 \]

Practice Problems

Problem 1: Compute:

\[ \frac{d}{dx} \left( x^5 - 7x^3 + 2x \right) \]

Solution:

\[ f'(x) = 5x^4 - 21x^2 + 2 \]

Problem 2: Compute:

\[ \frac{d}{dx} \left( (2x^2 - 1)(x^3 + 4) \right) \]

Solution (Product Rule):

\[ \frac{d}{dx} = (4x)(x^3 + 4) + (2x^2 - 1)(3x^2) \] \[ = 4x(x^3 + 4) + 3x^2(2x^2 - 1) \] \[ = 4x^4 + 16x + 6x^4 - 3x^2 \] \[ = 10x^4 - 3x^2 + 16x \]

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