Chain rule, implicit differentiation, inverse functions and higher derivatives

Learning objectives

Decompose a composite function into outer and inner parts and differentiate it with the chain rule, including compositions of three functions and combinations with the product and quotient rules

Differentiate exponentials and logarithms to any base using d/dx(aˣ) = aˣ ln a and d/dx(log_a x) = 1/(x ln a)

Differentiate a relation defined implicitly, solve the result for dy/dx, and use it to find tangent slopes and the points where the tangent is horizontal or vertical

Find the derivative of an inverse function at a point from (f⁻¹)′(b) = 1/f′(a) where f(a) = b, and explain why that formula follows from the chain rule

Recall and apply the derivatives of arcsin x, arccos x and arctan x, combining them with the chain rule

Compute second and higher-order derivatives of explicit functions and of implicit relations, and interpret f″ as the rate of change of f′