COALITION MATH 151, FALL 1997

        Group II (S. A. Fulling, assisted by Cary Lasher)

                              Day 7.1


    1.  The chain rule  (The rule for finding the derivative of a 
    composite function -- the powerful culmination of the 
    differentiation rules)   

    NOTE:  Use different color pens for inner and outer functions 
    and their contributions.

        A.  Example of use: d/dx [(x^2 + 1)^{-2}].  Think in 
        terms of y = u^{-2} with u = x^2 + 1.

        B.  General statements:

            i.  d/dx [f(g(x))] = f'(g(x)) g'(x) ;  
    
                (fog)' = (f'og) g' . 

            ii.  If y = f(u) and u = g(x), then 

                 dy/dx = dy/du du/dx .

        C.  (ii) is "obvious" and easy to remember.  It is 
        important to remember that dy/du is a function of u, so 
        u(x) must be substituted for its argument. 

        D.  Class exercises by teams mod 4:

            i.  d/dx (x^3 + 3x)^{1000}

            ii.  d/dt cos(\omega t + \theta)

            iii.  d/dx [(sin x)^3 + 3(sin x)^2]

            iv.  d/dx \sqrt{sin(\sqrt{x})}
THIS IS AN IMPROVEMENT OVER MY ORIGINAL CHOICE FOR iv.

        E.  A longer chain of functions:  (Use 4 colors.)

            d/dx {[sin(2x^2 + \sqrt{x^3})]^2}

        f(x) = x^2, g(x) = sin x, h(x) = 2x^2, k(x) = \sqrt x, 
        l(x) = x^3 => The structure is

            d/dx {f(g(h(x) + k(l(x))))}

        F.  Antiderivative of  sin(5x - 2), etc.
RAN OUT OF TIME HERE.

        G.  Do the postponed Part 2C of Day 6.2 

    2.  If time permits, discuss nonuniqueness in the polar angle 
    theta.

        A.  Undefined for r = 0.

        B.  Defined mod 2Pi elsewhere.

        C.  Any convention resolving the ambiguity produces a 
        discontinuity at some arbitrary place.  Favorite choices 
        are the positive and the negative horizontal axes.