Chain rule, Implicit Differentiation, Linear and
                   Quadratic Approximation

                                Mathematica

Dr. Andrzej Korzeniowski
Department of Mathematics, UTA

Lesson 3

Chain Rule

Suppose one wants to use the chain rule to differentiate h[x] = [Graphics:Images/lesson3_gr_1.gif] .  To this end define the outer function g[x]
and the inner function f [x] such that g [f [x]] = [Graphics:Images/lesson3_gr_2.gif].

[Graphics:Images/lesson3_gr_3.gif]
[Graphics:Images/lesson3_gr_4.gif]

[Graphics:Images/lesson3_gr_5.gif]

[Graphics:Images/lesson3_gr_6.gif]
[Graphics:Images/lesson3_gr_7.gif]

which is verified by  direct differentiation

[Graphics:Images/lesson3_gr_8.gif]
[Graphics:Images/lesson3_gr_9.gif]

Implicit Plot  -  Implicit Differentiation

[Graphics:Images/lesson3_gr_10.gif]
[Graphics:Images/lesson3_gr_11.gif]

[Graphics:Images/lesson3_gr_12.gif]

[Graphics:Images/lesson3_gr_13.gif]

DisplayFunction -> Identity delays rendering the graph until DisplayFunction -> $DisplayFunction

[Graphics:Images/lesson3_gr_14.gif]

[Graphics:Images/lesson3_gr_15.gif]

IMPLICIT  DIFFERENTIATION.   To differentiate implicitly replace y  by  y[x] and differentiate in x  as follows:

[Graphics:Images/lesson3_gr_16.gif]
[Graphics:Images/lesson3_gr_17.gif]
[Graphics:Images/lesson3_gr_18.gif]
[Graphics:Images/lesson3_gr_19.gif]
[Graphics:Images/lesson3_gr_20.gif]
[Graphics:Images/lesson3_gr_21.gif]
[Graphics:Images/lesson3_gr_22.gif]

Once y ' [x] is determined replace y[x] back by  y  in the above formula, i.e.,  y ' = [Graphics:Images/lesson3_gr_23.gif]

To find the slope of the tangent line at point (0,2) evaluate

[Graphics:Images/lesson3_gr_24.gif]
[Graphics:Images/lesson3_gr_25.gif]
[Graphics:Images/lesson3_gr_26.gif]
[Graphics:Images/lesson3_gr_27.gif]
[Graphics:Images/lesson3_gr_28.gif]

[Graphics:Images/lesson3_gr_29.gif]

Linear and Quadratic Approximation

The best way  to explain this concept is to sketch the graph of the function f [x],  tangent line and tangent parabola at a given point  x = a.    Approximations are as follows: L[x] =  f [a] + f '[a] (x-a)   (Linear),
Q[x] = f [a] + f '[a] (x-a) + [Graphics:Images/lesson3_gr_30.gif]f ''[a] [Graphics:Images/lesson3_gr_31.gif]   (Quadratic).

[Graphics:Images/lesson3_gr_32.gif]
[Graphics:Images/lesson3_gr_33.gif]
[Graphics:Images/lesson3_gr_34.gif]
[Graphics:Images/lesson3_gr_35.gif]
[Graphics:Images/lesson3_gr_36.gif]
[Graphics:Images/lesson3_gr_37.gif]

[Graphics:Images/lesson3_gr_38.gif]

Notice that the function f is plotted using default option, the function L using Dashing option, and the
function Q using the Thickness and GrayLevel options.  
In other words, plotting styles correspond to the order in which the functions are listed.


Converted by Mathematica      March 27, 2001