Differentiable functions!
I have a few questions:
If the directional derivatives $D_uf(a)$ exist for all directions $u$ and depend linearly on $u$, must $f$ be differentiable at $a$?
Also, how do I show that the function
$f: mathbb R^3 to mathbb R^2$
$f(x, y, z) = (e^{x+y+z},$cos x2y) is everywhere differentiable without making use of partial
derivatives?
analysis multivariable-calculus derivatives
edited Dec 2 at 17:09
mrtaurho
3,66621133
asked Dec 2 at 17:06
MathematicianP
2916
add a comment |
I have a few questions:
If the directional derivatives $D_uf(a)$ exist for all directions $u$ and depend linearly on $u$, must $f$ be differentiable at $a$?
Also, how do I show that the function
$f: mathbb R^3 to mathbb R^2$
$f(x, y, z) = (e^{x+y+z},$cos x2y) is everywhere differentiable without making use of partial
derivatives?
analysis multivariable-calculus derivatives
edited Dec 2 at 17:09
mrtaurho
3,66621133
asked Dec 2 at 17:06
MathematicianP
2916
add a comment |
I have a few questions:
If the directional derivatives $D_uf(a)$ exist for all directions $u$ and depend linearly on $u$, must $f$ be differentiable at $a$?
Also, how do I show that the function
$f: mathbb R^3 to mathbb R^2$
$f(x, y, z) = (e^{x+y+z},$cos x2y) is everywhere differentiable without making use of partial
derivatives?
analysis multivariable-calculus derivatives
edited Dec 2 at 17:09
mrtaurho
3,66621133
asked Dec 2 at 17:06
MathematicianP
2916
I have a few questions:
If the directional derivatives $D_uf(a)$ exist for all directions $u$ and depend linearly on $u$, must $f$ be differentiable at $a$?
Also, how do I show that the function
$f: mathbb R^3 to mathbb R^2$
$f(x, y, z) = (e^{x+y+z},$cos x2y) is everywhere differentiable without making use of partial
derivatives?
analysis multivariable-calculus derivatives
analysis multivariable-calculus derivatives
edited Dec 2 at 17:09
mrtaurho
3,66621133
asked Dec 2 at 17:06
MathematicianP
2916
edited Dec 2 at 17:09
mrtaurho
3,66621133
asked Dec 2 at 17:06
MathematicianP
2916
edited Dec 2 at 17:09
mrtaurho
3,66621133
edited Dec 2 at 17:09
mrtaurho
3,66621133
edited Dec 2 at 17:09
mrtaurho
3,66621133
3,66621133
asked Dec 2 at 17:06
MathematicianP
2916
asked Dec 2 at 17:06
MathematicianP
2916
asked Dec 2 at 17:06
MathematicianP
2916
2916
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Hint:
For the first part, what does it mean if the directional derivative exists in every direction? What about the linear aspect?
For the second part of your question, are the exponential function and the cosine function differentiable? What about if you compose a differentiable function with another, is it still differentiable?
answered Dec 2 at 17:19
user458276
12719
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
more hints please?
– MathematicianP
Dec 2 at 17:29
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint:
For the first part, what does it mean if the directional derivative exists in every direction? What about the linear aspect?
For the second part of your question, are the exponential function and the cosine function differentiable? What about if you compose a differentiable function with another, is it still differentiable?
answered Dec 2 at 17:19
user458276
12719
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
more hints please?
– MathematicianP
Dec 2 at 17:29
add a comment |
Hint:
For the first part, what does it mean if the directional derivative exists in every direction? What about the linear aspect?
For the second part of your question, are the exponential function and the cosine function differentiable? What about if you compose a differentiable function with another, is it still differentiable?
answered Dec 2 at 17:19
user458276
12719
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
more hints please?
– MathematicianP
Dec 2 at 17:29
add a comment |
Hint:
For the first part, what does it mean if the directional derivative exists in every direction? What about the linear aspect?
For the second part of your question, are the exponential function and the cosine function differentiable? What about if you compose a differentiable function with another, is it still differentiable?
answered Dec 2 at 17:19
user458276
12719
Hint:
For the first part, what does it mean if the directional derivative exists in every direction? What about the linear aspect?
For the second part of your question, are the exponential function and the cosine function differentiable? What about if you compose a differentiable function with another, is it still differentiable?
answered Dec 2 at 17:19
user458276
12719
answered Dec 2 at 17:19
user458276
12719
answered Dec 2 at 17:19
user458276
12719
answered Dec 2 at 17:19
user458276
12719
12719
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
more hints please?
– MathematicianP
Dec 2 at 17:29
add a comment |
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
more hints please?
– MathematicianP
Dec 2 at 17:29
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
I know that a linear map is always differentiable, so if all directional derivatives exist and are linear , then they are all differentiable?
– MathematicianP
Dec 2 at 17:21
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
This looks like a homework problem, so I’m only going to provide hints.
– user458276
Dec 2 at 17:22
more hints please?
– MathematicianP
Dec 2 at 17:29
more hints please?
– MathematicianP
Dec 2 at 17:29
add a comment |
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