Chain rule on two functions

The  exercise appears under the Differential calculus Math Mission. This exercise practices using the chain rule with two functions.

Types of Problems
There are two types of problems in this exercise:


 * 1) Use the graph and table to apply the chain rule: This problem provides a graph and a table for various functions. The student is asked to apply the chain rule, consulting the table and chart as relevant and necessary.Cro2f1.png
 * 2) Apply the chain rule directly: This problem provides a more direct application of the chain rule. The student is asked to find the correct output of the derivative at a particular value and indicate that answer.Cro2f2.png

Strategies
Knowledge of the chain rule and other differentiation rules are encouraged to ensure success on this exercise.
 * 1) In function notation the chain rule is (f(g(x))'=f'(g(x)*g'(x).
 * 2) In differential notation the chain rule is dy/dx=dy/du*du/dx.
 * 3) The chain rule can be thought of as the derivative of the big function at the inner function, multiplied by the derivative of the inner function.

Real-life Applications

 * 1) The derivative attempts to extend the concept of slope to objects that are not lines. It expresses a rate of change.
 * 2) Differential calculus applies to anything that changes or moves over time.
 * 3) The chain rule is the most foundational of all derivative rules as it can be seen to apply to all other rules and applications in the calculus.