Overview
This lesson covers the basics of derivatives, including rules for differentiating constants, monomials, polynomials, rational and radical functions, and introduces the product and quotient rules, as well as derivatives of trigonometric functions.
Derivatives of Constants and Monomials
- The derivative of any constant is zero.
- The derivative represents the slope of a function at a specific x-value.
- For f(x) = xⁿ, the power rule: derivative is n·x^(n-1).
Examples Using the Power Rule
- The derivative of x² is 2x.
- The derivative of x³ is 3x²; x⁴ is 4x³; x⁵ is 5x⁴.
- For constants multiplied by monomials, bring out the constant and apply the power rule.
Constant Multiple Rule
- The derivative of c·f(x) is c·(derivative of f(x)), where c is a constant.
Derivatives of Polynomials
- Differentiate each term separately using the power and constant multiple rules.
- Example: For f(x) = x³ + 7x² - 8x + 6, the derivative is 3x² + 14x - 8.
Finding Slopes and Interpretations
- The derivative at a point gives the slope of the tangent line at that x.
- The tangent line touches the curve at one point; the secant line at two points.
- Slope of a secant line approximates the slope of the tangent as points get closer.
Derivative by Definition (Limit Process)
- The derivative can be found with the limit: f'(x) = limₕ→₀ [f(x+h) - f(x)]/h.
Derivatives of Rational and Radical Functions
- Rewrite x in denominators using negative exponents before applying the power rule.
- For roots, convert to rational exponents, then apply the power rule, and rewrite as radicals if needed.
Product Rule
- For f(x)·g(x), derivative is f'(x)·g(x) + f(x)·g'(x).
- For three multiplied functions: differentiate each factor once, others stay the same, then sum.
Quotient Rule
- For f(x)/g(x), derivative is [g(x)·f'(x) - f(x)·g'(x)] / [g(x)]².
Derivatives of Trigonometric Functions
- The derivative of sin(x) is cos(x).
- The derivative of cos(x) is -sin(x).
- The derivative of tan(x) is sec²(x).
- The derivative of cot(x) is -csc²(x).
- The derivative of sec(x) is sec(x)tan(x).
- The derivative of csc(x) is -csc(x)cot(x).
Simplifying Before Differentiating
- Expand or simplify algebraic expressions before taking their derivatives when possible.
Key Terms & Definitions
- Derivative — A function giving the slope of a curve at any point.
- Power Rule — Shortcut for differentiating xⁿ: n·x^(n-1).
- Constant Multiple Rule — Derivative of constant times function: constant times the function's derivative.
- Product Rule — Rule for derivative of a product: f'(x)·g(x) + f(x)·g'(x).
- Quotient Rule — Rule for derivative of a quotient: [g(x)·f'(x) - f(x)·g'(x)] / [g(x)]².
- Tangent Line — Line that touches a curve at exactly one point.
- Secant Line — Line intersecting a curve at two points.
Action Items / Next Steps
- Practice differentiating various types of functions (monomials, polynomials, rational, radicals, trigonometric).
- Memorize the derivatives of basic trigonometric functions.
- Prepare for more advanced rules, such as the chain rule, in future lessons.