2. The Chain Rule

3. The Chain Rule is used to differentiate Composite Functions

4. What is your favorite song?... A song is an example of a composition.

5. How does a musician compose a song? drumspianovocalguitar

6. Engineers, architects, and graphic designers are composers as well.

10. Skateparks

13. Composite shapes are used all the time in art and graphic design.

14. Video game design..

15. Superheros…

17. Composite Functions When a function is composed of an inner function and an outer function, it is called a “composite function” When a relationship can be described using 2 functions.

18. The elevation (m) of an airplane is a function of time (s) The % of 0 2 in the atmosphere is a function of elevation t e 1 2 3 100 200 300 % 0 2 20 19.5 19

19. t e 1 2 3 100 200 300 % 0 2 20 19.5 19 We could then say the % of O 2 is a function of time… The polynomial equation that related these variables would be a “composite” function

20. WhalesWhales example

21. To differentiate a composite function, we use The Chain Rule

22. Differentiate f(x) = x 3 f’(x) = 3x 2 What if “x” was actually not just a variable, but another function altogether

23. Differentiate f(x) = (2x + 1) 3 The chain rule states 1. Differentiate the outer function first f’(x) = 3(2x + 1) 2 2. Differentiate the inner function second f’(x) = 3(2x + 1) 2 (2) f’(x) = 6(2x + 1) 2 3. Simplify

24. The Chain Rule Given two differentiable functions g(x) and h(x), the derivative of the composite function f(x) = g[h(x)] is f’(x) = g’[h(x)] X h’(x)

25. Determine the equation of the tangent to f(x) = 3x(1-x) 2 y = -0.75x + 0.75

26. Pg 117 3,4,7,8 For 8, read example 2 9,11,13,14,17