# Advanced differentiation

Contents

The chain rule sets the stage for implicit differentiation, which in turn allows us to differentiate inverse functions (and specifically the inverse trigonometric functions). This is really the top of the line when it comes to differentiation.

See how you score on these 20 practice questions

Some two-variable relationships cannot be turned into a function, like the circle equation x²+y²=4. Implicit differentiation allows us to find the derivative of y with respect to x, even in such equations.

Tackle some more advanced problems of implicit differentiation. For example, find dy/dx for e^(xy²)=x-y.

Implicit differentiation allows us to find the derivatives of arcsin(x), arccos(x), and arctan(x).

Implicit differentiation provides us with the relationship between the derivatives of inverse functions: if f and g are inverse functions, then f’(x)=1/(g’(f(x)). Get comfortable working with this relationship.

Equipped with knowledge about the derivatives of all common functions, evaluate some limits that represent various derivatives.

The derivatives of eˣ and ln(x) may be simple, but proving them is a different story. Learn all about it here.

In logarithmic differentiation, we find the derivative of the natural log of a function instead of the derivative of the function itself. It may surprise you, but this can sometimes actually be easier than regular differentiation, if not the only available option. For example, finding the derivative of xˣ.

Parametric and vector-valued functions basically take one input and return two outputs. How do we differentiate such functions? Find out here.

Review your knowledge of the advanced differentiation topics with some challenge problems.

The derivative is a function, and as such it has its own derivative! The same goes for the derivative of the derivative, and so forth. These are all called higher-order derivatives.

Learn how to find the second derivatives (and any higher-order derivative) of parametric and vector-valued functions.