implicit function + derivative definition ideas

implicit function + derivative definition ideas

@ (Calculus)

There is an impulse to see the equation is to solve the equation of expression, and then according to test instructions is to calculate the integral or the derivative, thought that is the solution.

Admittedly, the problem of solving an expression is, of course, the first to be solved. But when the feeling is not going down, to rational judgment, this problem should not be so, there should be more technical ideas.

To give an example:
(2013.9) Set function y = f (x) determined by Equation Y−x=ex (1−y) y-x = E^{x (1-y)}, Limn→∞n[f (1n) −1]=? \lim_{n\rightarrow \infty} n[f (\frac{1}{n})-1] =?

Analysis: A good idea is to first look at the solution of the problem form, which is easily lalacheche into a derivative of the definition form.

Limn→∞n[f (1n) −1]=limn→∞f (1n) −11n \lim_{n\rightarrow \infty} n[f (\frac{1}{n}) -1]\\ = \lim_{n\rightarrow \infty} \frac{ F (\frac{1}{n}) -1}{\frac{1}{n}}

Visible is the solution f′ (0) F ' (0), if the bold guess F (0) =1 f (0) = 1 do not have to solve the problem. Can save a little time. Also quickly, substituting for x=0, immediately get y=1.

Then the implicit function can be derivative once.

Y′=1=ex (1−y) −xy′→y′ (0) =1. Y ' = 1 = e^{x (1-y)-xy '} \rightarrow y ' (0) = 1. The END