Overview
This lecture covers key differentiation formulas in calculus, including rules for constants, powers, exponentials, products, quotients, the chain rule, logarithmic, trigonometric, and inverse trigonometric functions.
Basic Differentiation Rules
- The derivative of a constant is 0.
- The power rule: d/dx[xⁿ] = n·xⁿ⁻¹.
- The derivative of a constant raised to a variable: d/dx[aˣ] = aˣ·ln(a).
- For aˣ where x is a function, d/dx[aᵘ] = aᵘ·u'·ln(a).
- To differentiate xˣ, use logarithmic differentiation.
Constant Multiple, Product, and Quotient Rules
- Constant multiple rule: d/dx[c·f(x)] = c·f'(x).
- Product rule: d/dx[u·v] = u'·v + u·v'.
- Quotient rule: d/dx[u/v] = (v·u' - u·v') / v².
Chain Rule and Related Forms
- Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x).
- For f(x)ⁿ, derivative is n·f(x)ⁿ⁻¹·f'(x).
- dy/dx = dy/du · du/dx connects derivatives via substitution.
Logarithmic Differentiation
- d/dx[logₐ(u)] = u' / (u·ln(a)).
- d/dx[ln(u)] = u' / u.
Trigonometric Function Derivatives
- d/dx[sin(u)] = cos(u)·u'.
- d/dx[cos(u)] = -sin(u)·u'.
- d/dx[tan(u)] = sec²(u)·u'.
- d/dx[cot(u)] = -csc²(u)·u'.
- d/dx[sec(u)] = sec(u)·tan(u)·u'.
- d/dx[csc(u)] = -csc(u)·cot(u)·u'.
Inverse Trigonometric Function Derivatives
- d/dx[arcsin(u)] = u' / √(1 - u²).
- d/dx[arccos(u)] = -u' / √(1 - u²).
- d/dx[arctan(u)] = u' / (1 + u²).
- d/dx[arccot(u)] = -u' / (1 + u²).
- d/dx[arcsec(u)] = u' / [|u|·√(u² - 1)].
- d/dx[arccsc(u)] = -u' / [|u|·√(u² - 1)].
Key Terms & Definitions
- Derivative — Measures the instantaneous rate of change of a function.
- Constant multiple rule — The derivative of a constant times a function is the constant times the function’s derivative.
- Product rule — Formula to differentiate products of two functions.
- Quotient rule — Formula to differentiate ratios of two functions.
- Chain rule — Rule for differentiating compositions of functions.
Action Items / Next Steps
- Review and memorize these differentiation formulas.
- Practice problems on all types of differentiation rules.
- Watch extra videos or review textbook sections on logarithmic differentiation and the chain rule if needed.