##
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.**

Converted by *Mathematica*
March 27, 2001