You are here:

Calculus/derivatives

Advertisement

Expert: Alon Mandes - 9/25/2008


Question
if x^2 -y^2 = 1, find the second derivative at the value (2, square root of 3)

Answer
Ok Kristin , finding derivative of Implicit functions can be done
in 2 ways :

1st way : x^2-y^2=1 . It can be wrote as y=sqrt[x^2-1]. then the
derivative of y with respect to x will be : y'=(x/sqrt[x^2-1]).
To find y'' we simply derive y' with respect to x :
y''= { sqrt[x^2-1]-x^2/sqrt[x^2-1] }/{ x^2-1 }=-1/(x^2-1)^(3/2).
Note : [f(x)/g(x)]'={ f'(x)g(x)-g'(x)f(x) }/g(x)^2.
y''(at x=2)=-1/3^(3/2)=-0.192


2nd way : x^2-y^2=1 . We will derive with respect to x :
2x-2yy'=0 --> y'=x/y. For finding y'' we derive y' with respect
to x : y''=(y-xy')/y^2 --> y''=(y-x(x/y))/y^2=(y^2-x^2)/y^3
y''[2,sqrt(3)]=(3-4)/3^(1.5)=-0.192

Notice that we got the same answer.

Alon.  

Calculus

All Answers

Answers by Expert:

Ask Experts

Volunteer

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

©2012 About.com, a part of The New York Times Company. All rights reserved.