Section 3.1 - Basic Derivative Rules

3.1 Basic Derivative Rules

• Introduction to derivative rules based on the definition of a derivative.

Definition of the Derivative

• The Derivative of f at x is defined as:

  • f′(x) = lim(h→0) (f(x + h) − f(x)) / h

Derivative of Constant Functions

• Constant Function: f(x) = c

  • Derivative: f′(x) = 0

  • Example:

    • For f(x) = 3, f′(x) = 0

    • For f(x) = 4x,

  • Derivative of f(x) = mx (where m is real): f′(x) = m

Derivative of Linear Functions

• Linear Function: f(x) = mx + b (m and b are real numbers)

  • Derivative: f′(x) = m

Finding Derivatives of Specific Functions

• Example with f(x) = x:

  • Applying the derivative definition:

    • f′(x) = lim(h→0) (f(x + h) − f(x)) / h

    • Calculate to find f′(x) = 1

• Example with f(x) = x²:

  • Expansion yields: f′(x) = 2x

• Example with f(x) = x³:

  • Expansion results in: f′(x) = 3x²

Power Rule Development

• From the examples, we see a pattern:

  • Power Rule: If f(x) = xⁿ (n is any real number), then f′(x) = n*x^(n−1)

• Common Expression: d/dx (xⁿ) = n*x^(n−1)

• Example: Derivative of f(x) = 1/x⁴:

  • Write as x⁻⁴, apply power rule: f′(x) = -4*x⁻⁵ = -4/x⁵

Derivatives of Polynomials

• Coefficients remain unchanged during differentiation:

  • Derivative of a Constant Times a Function: If f(x) = kg(x), then f′(x) = kg′(x)

• Example:

  • Find derivatives:

    • (a) f(x) = 5x⁹

    • (b) f(x) = -2x⁻³

    • (c) f(x) = 7x⁶

Sum and Difference Derivatives

• Derivative of the Sum of Two Functions: f(x) = g(x) + h(x)

  • f′(x) = g′(x) + h′(x)

• Derivative of the Difference of Two Functions: f(x) = g(x) - h(x)

  • f′(x) = g′(x) - h′(x)

• Proof: Derivative calculations using limit definitions result in the above rules.

• Examples presented:

  • (a) f(x) = 3x² + 5x

  • (b) f(x) = 7x⁵ - 4x⁴ + 3x³ - 6x² + x - 5

Further Derivative Examples

• Continued examples include derivatives of quadratic, cubic, and polynomial functions:

  • (a) f(x) = -x² - 3x - 6

  • (b) f(x) = (1/3)x³ - (1/2)x² + 5

  • (c) f(x) = 4x³ - 3x²

• Includes more advanced functions like products and roots:

  • Find examples like:

    • (a) f(x) = 2x⁵ + 3x⁴ - 7x³

    • (b) f(x) = √x

Tangent Line Example Problem

• Given f(x) = -2x³ + 6x - 8, find f′(2) to interpret its meaning.

• Examining the tangent line to f(x) = 4x² - 3x at x = -1.