Overview
This lecture covers the fundamentals of differentiation, including rules for finding derivatives of constants, monomials, polynomials, rational and radical functions, as well as the derivatives of trigonometric functions and the key rules: product and quotient.
The Concept of the Derivative
- The derivative of a constant is 0.
- The derivative gives the slope of a function at any x value.
- Derivative notation: d/dx or f'(x).
Power Rule and Constant Multiple Rule
- Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹.
- Constant Multiple Rule: d/dx[c·f(x)] = c·d/dx[f(x)].
- Example: d/dx[4x⁷] = 28x⁶.
Derivative Examples
- d/dx[x²] = 2x.
- d/dx[x³] = 3x²; d/dx[x⁴] = 4x³; d/dx[x⁵] = 5x⁴.
- d/dx[8x⁴] = 32x³; d/dx[5x⁶] = 30x⁵.
Derivative Using First Principles
- Definition: f'(x) = lim_{h→0} [f(x+h) - f(x)] / h.
- Example with f(x) = x² confirms d/dx[x²] = 2x.
Tangent vs. Secant Lines
- Tangent line: touches curve at one point; its slope is given by the derivative.
- Secant line: crosses the curve at two points; its slope approximates the tangent slope as points get closer.
Derivatives of Polynomials
- Differentiate each monomial term separately.
- Example: d/dx[x³ + 7x² - 8x + 6] = 3x² + 14x - 8.
Derivatives of Rational Functions
- Rewrite fractions with exponents (e.g., x⁻¹).
- Apply the power rule to negative exponents.
- Example: d/dx[1/x²] = -2/x³.
Derivatives of Radical Functions
- Rewrite radicals with fractional exponents (e.g., √x = x¹/²).
- Apply the power rule.
- Example: d/dx[√x] = 1/(2√x).
Simplifying Before Differentiating
- Expand or simplify expressions before differentiating.
- Divide each term by denominators if needed before applying rules.
Derivatives of Trigonometric Functions
- d/dx[sin x] = cos x.
- d/dx[cos x] = -sin x.
- d/dx[tan x] = sec² x.
- d/dx[sec x] = sec x·tan x.
- d/dx[csc x] = -csc x·cot x.
- d/dx[cot x] = -csc² x.
Product Rule
- If y = f(x)·g(x), then y' = f'(x)·g(x) + f(x)·g'(x).
- For three functions: differentiate each, keeping others the same, then sum.
Quotient Rule
- If y = f(x)/g(x), then y' = [g(x)·f'(x) - f(x)·g'(x)] / [g(x)]².
Key Terms & Definitions
- Derivative — The function giving the slope at each point of another function.
- Monomial — A single term algebraic expression.
- Power Rule — Shortcut for differentiating xⁿ.
- Constant Multiple Rule — Allows constants to be factored out before differentiating.
- Secant Line — Line intersecting a curve at two points.
- Tangent Line — Line touching a curve at only one point.
- Product Rule — Formula for the derivative of a product of functions.
- Quotient Rule — Formula for the derivative of a quotient of functions.
Action Items / Next Steps
- Practice differentiating polynomials, rational, and radical functions.
- Memorize basic trigonometric derivatives.
- Complete homework on applying product and quotient rules.