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Calculus/Calculus - Derivatives

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Expert: Alon Mandes - 10/3/2009


Question
Hello Mr. Mandes,

I need to find the derivative of f(x) = 1 / (squareroot) of (x+2).
I've been working on this question for about 30 minutes now and I can't figure out how to do it. I can do it using the shortcut (obviously) but using the f(prime)(x)= f(x+h) - f(x) / h, I only get about 3 steps in and get confused.

Any help would be much appreciated.
Thanks,
Francis

Answer
Hello Francis,
In this section we will use the trick : √a-√(a+b)=-b/[√a+√(a+b)]. This is true because :
√a-√(a+b)=√a-√(a+b)*[√a+√(a+b)]/[√a+√(a+b)]=a-a+b/[√a+√(a+b)]=-b/[√a+√(a+b)].
So,
Lim  [f(x+h)-f(x)]/h = f'(x)
h->0       
In our case f(x)=1/√(x+2) .
Thus,
Lim   [1/√(x+2+h) - 1/√(x+2)]/h =
h->0
Lim   [√(x+2) - √(x+2+h)]/[√(x+2+h)√(x+2)]h =
h->0
Lim   -h/[√(x+2+h)√(x+2)][√(x+2)+√(x+2+h)]h =
h->0
Lim   -1/[√(x+2+h)√(x+2)][√(x+2)+√(x+2+h)]
h->0
As h goes to zero we get :
-1/[√(x+2+0)√(x+2)][√(x+2)+√(x+2+0)]=-1/[√(x+2)√(x+2)][√(x+2)+√(x+2)]=
-1/[(x+2)²][2√(x+2)]=-1/[2(x+2)^(3/2)]=f'(x) .

Alon.  

Calculus

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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".

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

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

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