Chain rule, Implicit Differentiation, Linear and
                   Quadratic Approximation


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].




which is verified by  direct differentiation


Implicit Plot  -  Implicit Differentiation




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



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


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



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).



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