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] = .  To this end define the outer function g[x]
and the inner function f [x] such that g [f [x]] = .

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 ' =

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) + f ''[a]    (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.