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Mano Jow
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What's the condition for f(x,y) to be differentiable in its domain?
I googled for it but couln't find...
Thanks in advance.
I googled for it but couln't find...
Thanks in advance.
No. A function [itex]x\mapsto f(x)[/itex] is differentiabe at [itex]x_0[/itex] if the limitMano Jow said:For a function of one argument to be differentiable it needs to be countinous in its domain?
Mano Jow said:Uhmmm the question is:
"Verify if each function is differentiable at the points of its domain."
a)f(x,y) = 2x - y
b)f(x,y) = ln(2x²+3y²)
Mark44 said:Do you know what the domain is for each of these functions?
Pere Callahan said:Certainly, if you are asked to prove differentiability of these two functions you are given some sort of definition of this concept.
I suggest you review this definition and then we can see where you might have problems verifying it for the two examples you gave.
Yes, as far as it goes. For the second function, how else could you describe the domain?Mano Jow said:For the first one, the domain is R², right? And for the second, as there is no log of numbers equal or below zero, we have that x²+y²>0. Isn't that right?
Mano Jow said:The teacher told us to use a certain book to study, but this is not there...
Thanks again.
HallsofIvy said:"F(x,y) is differentiable at [itex](x_0, y_0)[/itex] if and only if the partial derivatives [itex]\partial F/\partial x[/itex] and [itex]\partial F/\partial y[/itex] exist and are continuous in some neighbohood of [itex](x_0, y_0)[/itex]."
A function is differentiable in R2 if it is differentiable at each point in R2.Mano Jow said:But how can I find this [itex](x_0, y_0)[/itex] when the exercise asks if its differentiable in R²?
I know what you're trying to say, but you're not saying it very well. For this problem, it's easier to say where it is not defined.Mano Jow said:For the second function, I can also say that the domain is all the circunferences with a radius bigger than 0... does it helps in anything?
Partial derivatives. There are two of them.Mano Jow said:Perhaps I figured that out for the second one:
I should find the partial derivative of the function using limits when x and y tend to 0?
But that's the easier one!Mano Jow said:Yet the first one isn't very clear to me...
Differentiability is a mathematical concept that describes the smoothness of a function. A function is said to be differentiable at a point if its derivative exists at that point. In other words, the function has a well-defined slope at that point.
Continuity and differentiability are related but distinct concepts. Continuity describes the smoothness of a function, while differentiability describes the existence of a derivative. A function can be continuous at a point without being differentiable at that point, but if a function is differentiable at a point, it must also be continuous at that point.
The geometric interpretation of differentiability is that a differentiable function is locally linear at a given point. This means that the graph of the function can be approximated by a straight line with a well-defined slope at that point.
In order for a function to be differentiable at a point, it must be continuous at that point and have a well-defined slope (i.e. derivative) at that point. Additionally, the left and right-hand limits of the function must be equal at that point.
Yes, a function can be differentiable at a single point but not on an interval. This can occur if the function is not continuous on that interval or if the derivative does not exist at certain points within the interval.