1. Chain Rule
Chain Rule

2. Objectives Students will be able to
Apply the chain rule to find derivatives.
Chain Rule

3. Rules
Product Rule
If g is differentiable at x0 and f is differentiable at u0 = g(x0) then is differentiable at x0 and the derivative is
Chain Rule

4. Example 1
Let and Find
f[g(2)] and g[f(2)]
Chain Rule

5. Example 2
Let and Find
f[g(x)] and g[f(x)]
Chain Rule

6. Example 3
Find the derivative of
Chain Rule

7. Example 4
Find the derivative of
Chain Rule

8. Example 5
Find the derivative of
Chain Rule

9. Example 6-1
Suppose the cost in dollars of manufacturing q items is given by
and the demand equation is given by
In terms of the demand
Find an expression for the revenue R.

10. Example 6-2
Suppose the cost in dollars of manufacturing q items is given by
and the demand equation is given by
In terms of the demand
Find an expression for the profit P.

11. Example 6-3
Suppose the cost in dollars of manufacturing q items is given by
and the demand equation is given by
In terms of the demand
Find an expression for the marginal profit.

12. Example 6-4
Suppose the cost in dollars of manufacturing q items is given by
and the demand equation is given by
In terms of the demand
Determine the value of marginal profit when the price is $5000.