1. Winter wk 4 – Tues.25.Jan.05 Review:
Polynomial rule for derivatives
Differentiating exponential functions
Higher order derivatives
How to differentiate combinations of functions?
Product rule (3.3)
Quotient rule (3.4)
Energy Systems, EJZ

2. Differentiating polynomials and ex
Integrating polynomials:
Slope of ex increases exponentially:
d/dx(ex) = ex
d/dx(ax) = ln(a) ax

3. Higher order derivatives
Second derivative = rate of change of first derivative
f ’<0
f ’>0
f ’’>0
f ’>0
f ’<0
f ’’<0

4. Ch.3.3: Products of functions
If these are plots of f(x) and g(x)
Then sketch the product y(x) = f(x).g(x) = f.g

5. Differentiating products of functions
Ex: We couldn’t do all derivatives with last week’s rules: y(x) = x ex. What is dy/dx?
Write y(x) = f(x) g(x).
Slope of y = (f * slope of g) + (g * slope of f)
Try this for y(x) = x ex, where f=x, g=ex

6. Proof (justification)

8. Spend 10-15 minutes doing odd # problems on p.121
Practice – Ch.3.3
Spend minutes doing odd # problems on p.121
Pick one or two of these to set up together:
32, 38, 45

9. Ch.3.4 Functions of functions
Ex: We couldn’t do 3.1 #36 with last week’s rules: y =(x+3)½ What is dy/dx?
Consider y(x) = f(g(x)) = f(z),
where z=g(x).
Try this for y =(x+3)½ , where z=x+3, f=z½

10. Proof (justification)
Differentiating functions of functions: y(x) = f(g(x))
See #17, p.154
Derive the chain rule using local linearizations:
g(x+h) ~ g(x) + g’(x) h =
f(z+k) ~ f(z) + f’(z) k =
y’ = f’(g(x)) =

11. Differentiating functions of Functions
If these are plots of f(x) and g(x)
Then sketch function y(x) = f(g(x)) = f(g)

12. Candidates for y= f(g(x))

13. Answer

14. Calc Ch.3.4 Conceptest 2

15. Calc Ch.3.4 Conceptest 2 options

16. Calc Ch.3.4 Conceptest 2 answer

17. Calc Ch.3.4 Conceptest 3

18. Calc Ch.3.4 Conceptest 3 options

19. Calc Ch.3.4 Conceptest 3 answer

20. Spend 10-15 minutes doing odd # problems on p.126
Practice – Ch.3.4
Spend minutes doing odd # problems on p.126
Pick one or two of these to set up together:
52, 54, 62, 66, 68