Definition: The derivative of arcsin$x$ equals 1/√$1-x²$, valid within the domain $-1,1$.

Highlight: Higher-order derivatives are denoted as f'$x$, f''$x$, f'''$x$, and f⁽ⁿ⁾$x$ for the nth derivative.

Example: For inverse trigonometric functions, arctan$x$ has a domain of all real numbers and range of $-π/2, π/2$.

Vocabulary: The notation dy/dx represents the first derivative, while d²y/dx² represents the second derivative.

Definition: The chain rule states that the derivative of a composite function f(g$x$) equals f'(g$x$) multiplied by g'$x$.

Example: When differentiating ln(x³), using the chain rule gives us 3/x.

Highlight: For implicit differentiation, remember to include dy/dx whenever differentiating terms containing y.

Vocabulary: Horizontal tangent lines occur when dy/dx = 0, while vertical tangent lines exist when dy/dx is undefined.