How to prove partial derivatives exist

[asif zaidi]

I am really struggling with this h/w problem...especially the 1st part.

Problem Statement:

Consider the function f defined by f(x1,x2,x3)=cos(x1+x2)+exp(sin(x1*x2*x3)+cos(x1^{2}+x3^{2})).

Show that the partial derivatives exist and are continuous everywhere.


Solution

1- I can find fx1(x1, x2, x3), fx2(x1, x2, x3) and fx3[/tex] (x1, x2, x3)

Does this mean that partial derivatives exist ?

Alternatively do I have to use the definition of partial derivatives as follows

lim (h->0) ( f(x1+ah, x2+bh, x3+ch) - f(x1, x2, x3) ) / h. If I do this, there is no way I can evaluate the function as given above.

Plz advise how to proceed?


2- To prove that it is continuous

cos(x1+x2) = cos(x1)cos(x2) - sin(x1)sin(x2). Sin and Cos are continuous functions. product of continuous functions is also continuous.

For other parts repeat same logic. Basically sum of continuous functions is also continuous.

Is this right approach.


Thansk

Asif

 

[Dick]

That's the right approach. Once you've learned to differentiate a function you don't need to go back to the difference quotient. And yes, once everything in sight is continuous and has no singularities for x1,x2 and x3 you can say the whole thing is continuous. No need to mess around.