Unit Three: Differentiation: Composite, Implicit, and Inverse Functions- essential knowledge

What does the chain rule provide in calculus?

The chain rule provides a way to differentiate composite functions.

What role does the chain rule play in implicit differentiation?

The chain rule is the basis for implicit differentiation.

How can the chain rule and the definition of an inverse function be used together?

They can be used to find the derivative of an inverse function, provided the derivative exists.

How are the derivatives of inverse trigonometric functions found?

The chain rule applied with the definition of an inverse function, or the formula for the derivative of an inverse function, can be used to find the derivatives of inverse trigonometric functions.

What does differentiating the first derivative of a function produce?

Differentiating f′ produces the second derivative f′′, provided the derivative of f′ exists; repeating this process produces higher-order derivatives of f.

How are higher-order derivatives represented in notation?

For y = f(x), notations for the second derivative include d²y/dx², f''(x), and y''. Higher-order derivatives can be denoted dⁿy/dxⁿ or fⁿ(x).