- #1
Chris L
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As a preface, this question is taken from Vector Calculus 4th Edition by Susan Jane Colley, section 2.3 exercises.
"Explain why each of the functions given in Exercises 34-36 is differentiable at every point in its domain."
34. [tex]xy - 7x^8y^2 cosx[/tex]
35. [tex]\frac{x + y + z}{x^2 + y^2 + z^2}[/tex]
36. [tex](\frac{xy^2}{x^2 + y^4}, \frac{x}{y} + \frac{y}{x})[/tex]
Not equations, rather theorems, but basically for a function to be differentiable at a point a the function itself must exist at a and each of its partial derivatives must be continuous at a.
It is immediately clear that 34 must be differentiable everywhere, no objections to this question. However, I have problems with 35 and 36:
35: First and foremost, the function isn't defined at (0,0,0) (as 0/0 isn't defined). If you compute its partial derivatives, they also are not defined at (0,0,0) (also equal to 0/0), so from the definitions in the book, the function isn't differentiable everywhere. Am I misinterpreting the question/have the wrong conditions for differentiability or is the question in the book worded poorly (perhaps the intended problem is to explain whether or not each function given is differentiable everywhere?)
36: Similar to 35, this function isn't defined when either x or y is 0, and its partial derivatives aren't either.
Thanks in advance.
Homework Statement
"Explain why each of the functions given in Exercises 34-36 is differentiable at every point in its domain."
34. [tex]xy - 7x^8y^2 cosx[/tex]
35. [tex]\frac{x + y + z}{x^2 + y^2 + z^2}[/tex]
36. [tex](\frac{xy^2}{x^2 + y^4}, \frac{x}{y} + \frac{y}{x})[/tex]
Homework Equations
Not equations, rather theorems, but basically for a function to be differentiable at a point a the function itself must exist at a and each of its partial derivatives must be continuous at a.
The Attempt at a Solution
It is immediately clear that 34 must be differentiable everywhere, no objections to this question. However, I have problems with 35 and 36:
35: First and foremost, the function isn't defined at (0,0,0) (as 0/0 isn't defined). If you compute its partial derivatives, they also are not defined at (0,0,0) (also equal to 0/0), so from the definitions in the book, the function isn't differentiable everywhere. Am I misinterpreting the question/have the wrong conditions for differentiability or is the question in the book worded poorly (perhaps the intended problem is to explain whether or not each function given is differentiable everywhere?)
36: Similar to 35, this function isn't defined when either x or y is 0, and its partial derivatives aren't either.
Thanks in advance.