1. 3.6
Chain Rule

2. Quick Review
In Exercises 1 –5, let Write a simplified expression for the composite functions.

3. Quick Review
In Exercises 6 –10, let Write the given function as a composite of two or more of f, g, and h. For example,

4. What you’ll learn about
Derivative of a Composite Function
“Outside-Inside” Rule
Repeated Use of the Chain Rule
Slopes of Parametrized Curves
Power Chain Rule
Essential Question
Why is the chain rule the most widely used differentiation rule in mathematics and how do we use it?

5. Rule 8 The Chain Rule
In Leibniz notation, if y = f (u) and u = g (x), then

6. Example Derivatives of Composite Functions
Suppose that the function f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x
f (x)
g (x)
f ‘ (x)
g‘ (x) 3 1 16 8 5 4 -3 2 -5

7. Example Derivatives of Composite Functions
We know the function Use this to show that the derivatives are equal using composition of functions.

8. “Outside-Inside” Rule

9. Example “Outside-Inside” Rule
A object moves along the x-axis so that its position at any time t > 0 is given by x (t) = cos(t ). Find the velocity of the object as a function of t.

10. Example “Outside-Inside” Rule
Find the derivatives of each.

11. Example “Outside-Inside” Rule
Find the derivatives of each.

12. Example “Outside-Inside” Rule
Find the value of at the given value of x.

13. Example Repeated Use of the Chain Rule

14. Slopes of Parametrized Curves
Finding dy/dx Parametrically

15. Example Finding dy/dx Parametrically
Find the equation of the line tangent to the curve defined parametrically by

16. Power Chain Rule

17. Example Using Chain Rule to Find Tangents
Find the slope of the line tangent to the curve at the point where x = p /3.

18. Example Using Chain Rule to Find Tangents
Show that the slope of every line tangent to the curve is positive.
Because the denominator is being raised to an even power, the slope will always have a positive value.

19. Use the given substitution and the Chain Rule to find dy/dx.

20. Find dy/dx.

21. Find y’’ .

22. Find the equation of the line tangent to the curve at the given point.

23. Pg. 92, 2.4 #2-40 even