Overview
This lecture covers the basics of finding derivatives, including rules for constants, monomials, polynomials, rational functions, radicals, trigonometric functions, and the product and quotient rules.
Derivatives of Constants and Monomials
- The derivative of any constant is zero (e.g., d/dx[5]=0).
- The derivative gives the slope of a function at a point.
- The power rule: d/dx[xⁿ] = n·xⁿ⁻¹.
- Examples: d/dx[x²] = 2x, d/dx[x³] = 3x², d/dx[x⁴] = 4x³.
- The constant multiple rule: d/dx[c·f(x)] = c·f'(x).
Derivative Definition and Applications
- The derivative can be defined as f'(x) = limₕ→₀ [f(x+h)-f(x)]/h.
- This limit shows the slope at a specific point (tangent line).
- The secant line connects two points; as these points approach each other, the secant slope approaches the tangent slope.
- The derivative at a point gives the slope of the tangent line at that x value.
Derivatives of Polynomials and Rational Functions
- For polynomials, differentiate each term separately using the power and constant multiple rules.
- Example: d/dx[x³ + 7x² - 8x + 6] = 3x² + 14x - 8.
- For rational functions like 1/x or 1/x², rewrite as x⁻¹ or x⁻², use the power rule, then simplify.
- Example: d/dx[1/x] = -1/x².
Derivatives of Radical Functions
- Rewrite roots as rational exponents, then apply the power rule.
- Example: d/dx[√x] = 1/(2√x); d/dx[∛(x⁵)] = 5x²/³ / 3.
Combining and Simplifying Before Differentiation
- Multiply out or simplify expressions before differentiating if possible.
- Divide terms individually when a polynomial is divided by a monomial.
Derivatives of Trigonometric Functions
- d/dx[sin x] = cos x.
- d/dx[cos x] = -sin x.
- d/dx[sec x] = sec x tan x.
- d/dx[csc x] = -csc x cot x.
- d/dx[tan x] = sec²x.
- d/dx[cot x] = -csc²x.
Product Rule
- For two functions: d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x).
- For three functions: Differentiate each part in turn, holding others constant in each term.
Quotient Rule
- For f(x)/g(x): d/dx[f/g] = (g·f' - f·g') / (g²).
Key Terms & Definitions
- Derivative — Slope of a function at a specific point.
- Power Rule — d/dx[xⁿ] = n·xⁿ⁻¹.
- Constant Multiple Rule — d/dx[c·f(x)] = c·d/dx[f(x)].
- Product Rule — d/dx[f·g] = f'·g + f·g'.
- Quotient Rule — d/dx[f/g] = (g·f' - f·g')/g².
- Secant Line — Intersects curve at two points.
- Tangent Line — Touches curve at only one point.
Action Items / Next Steps
- Practice differentiating polynomials, rational functions, and radicals.
- Memorize basic derivative formulas for trigonometric functions.
- Review and apply product and quotient rules to examples.
- Prepare for homework involving derivative calculations and simplification of functions before differentiating.