Overview
This lecture covers the basics of derivatives, including rules for finding derivatives of constants, monomials, polynomials, rational and radical functions, basic trigonometric derivatives, and introduces the product and quotient rules.
The Concept of the Derivative
- The derivative gives the slope of a function at any x-value.
- Derivative of a constant is zero.
- Notation: d/dx[f(x)] 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·f'(x).
- Examples: d/dx[x³] = 3x², d/dx[4x⁷] = 28x⁶.
Derivative of Polynomials
- Differentiate each term separately using the power and constant multiple rules.
- Example: d/dx[x³ + 7x² - 8x + 6] = 3x² + 14x - 8.
Derivative by Definition (Limit Process)
- f'(x) = lim_{h→0} [f(x+h) - f(x)] / h.
- Used to confirm the correctness of rules for specific functions like x²._
Tangent and Secant Lines
- The derivative gives the slope of the tangent line at a point.
- Secant line connects two points; as they approach each other, the secant slope approaches the tangent slope.
Derivatives of Rational and Radical Functions
- Rewrite 1/x as x⁻¹, then use power rule: d/dx[1/x] = -1/x².
- For radicals, write √x as x^{1/2}: d/dx[√x] = 1/(2√x).
Trigonometric Derivatives
- 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
- d/dx[f(x)·g(x)] = f'(x)g(x) + f(x)g'(x).
- Extended for three functions: Differentiate one at a time, keep others unchanged.
Quotient Rule
- d/dx[f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]².
Key Terms & Definitions
- Derivative — The instantaneous rate of change of a function with respect to x.
- Monomial — A term of the form cxⁿ.
- Power Rule — Formula for the derivative of x raised to a constant power.
- Constant Multiple Rule — The derivative of a constant times a function.
- Tangent Line — A line touching the curve at one point, matching its slope there.
- Secant Line — A line that passes through two points on a curve.
- Product Rule — Rule for differentiating the product of two or more functions.
- Quotient Rule — Rule for differentiating the quotient of two functions.
Action Items / Next Steps
- Practice differentiating polynomials, rational, and radical expressions.
- Memorize basic trigonometric derivatives.
- Review and apply the product and quotient rules in homework problems.
- Expand expressions or simplify before differentiating when necessary.