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
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
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.