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Calculus/Partial derivative, differentiability

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Expert: - 3/19/2009


Question
Hi, I am Ssa.
1. For a function f(x,y) to be differentiable, its partial derivatives have to be defined and continuous right?
2. How do you check if the partial derivatives are continuous? Same as checking continuity of a function?
I was given this question: f(x,y)=1 - sin(sqrt(x^2+y^2))
I can get the partial derivatives our but I'm not sure if this function is differentiable or not.

I appreciate ur help a lot. thanx

Answer

Surface
Questioner:   Ssa
Country:  Singapore
Category:  Calculus
Private:  No
 
Subject:  Partial derivative, differentiability
Question:  Hi, I am Ssa.
1. For a function f(x,y) to be differentiable, its partial derivatives have to be defined and continuous right?
2. How do you check if the partial derivatives are continuous? Same as checking continuity of a function?
I was given this question: f(x,y)=1 - sin(sqrt(x^2+y^2))
I can get the partial derivatives our but I'm not sure if this function is differentiable or not.

I appreciate ur help a lot. thanx
..........................................
Note: Please write English. I don't understand onlinesese.  What is a ur?

Now about derivatives:
                               x
f_x = - cos(sqrt(x^2+y^2)) --------------
                          sqrt(x^2+y^2)   

That would be defined and continuous except where that denominator is zero, which would be at  (0,0).

The same for
                                y
f_y = - cos(sqrt(x^2+y^2)) --------------
                          sqrt(x^2+y^2)   

(See attached surface graph.)

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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