Derivatives of inverse and inverse trigonometric functions, and higher-order derivatives

Learning objectives

Use the relationship (f⁻¹)′(x) = 1/f′(f⁻¹(x)) to find the derivative of an inverse function at a point from a formula, a table or a graph of f, and explain where it comes from using the chain rule

Recall and apply the derivatives of arcsin x, arccos x, arctan x and the other inverse trigonometric functions, including with the chain rule

Select and combine the appropriate differentiation rules for an expression, simplifying algebraically first when this makes the derivative easier

Find second and higher-order derivatives, using the notations f″(x), d²y/dx², y″ and f⁽ⁿ⁾(x), for explicit functions and for implicitly defined relations

Interpret a second derivative as the rate of change of the first derivative, and evaluate higher-order derivatives at a point from tables or graphs