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Calculus III

Course: Calculus I
Topic: Derivatives
Subtopic: Chain Rule

Overview

Functions such as sinsqrtx have an "inside" function, sqrtx, and an "outside" function, the sine function. Such functions are really a composition of two functions, e.g. f(x)=sinx, g(x)=sqrtx, h(x)=sin sqrtx=(f @ g)(x). The chain rule is used for finding derivatives of composed functions, functions where there is one function inside another.

When you have multiple composed functions, it is helpful to start with the very inside function and work your way out or visa versa. Get in the habit of going one direction and sticking to it else taking the derivative of something like y=sqrt(sin(e^cos(x^3))) becomes error prone!

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 2.5.1 Understand the chain rule using both Newton and Leibniz notations
• 2.5.2 Apply the chain rule to differentiate a composition of functions
• 2.5.3 Find slope and equation of tangent line to a composed function using the derivative without using the formal definition of derivative

Terminology

Terms you should be able to define: chain rule, inside vs. outside of a composed function

Text Notes

It is extremely important to be able to use the chain rule accurately. Practice, practice, practice!

Mini-Lectures and Examples

STUDY: Derivatives - Chain Rule

Supplemental Resources (optional)

Supplemental Resources (optional)

rev. 2020-10-10