Question

differentiate e^(x/y)-(x/y)e^(x/y) in term of x

Answer 1

differentiate in term of x \[e ^{x/y}(1-x/y) \]

Answer 2

partial derivative?

Answer 3

or implicit derivative? sorry, since i have two systems, i have to make sure what the problem is

Answer 4

ok, mybe u should write the full question, because this is involving exact equation

Answer 5

\[(1+e ^{x/y})dx + e ^{x/y}(1-x/y)dy = 0\]

Answer 6

and i couldn't figure how to differentiate the dy part

Answer 7

It's unclear at all, it seems implicitly derivative respect to x , how can you break the first term to x (dx) and second term to y (dy) like that? does your pro say that?

Answer 8

no, this is exactly what the question is ._.

Answer 9

moreover, take integral, we have dx, dy. take derivative, we don't have them

Answer 10

inside the book ._.

Answer 11

lemme snap a pic, n show

Answer 12

3a

Answer 13

first check if they are exact. If they are not exact, you don't need to proceed, but you CAN make them exact by finding an integrating factor.

Answer 14

find: \[\psi_x, \psi_y\]

Answer 15

yeah, but i couldn't find it since idk how to differentiate the dy part @.@

Answer 16

\[\frac{ ∂ }{ ∂y }(1+e^{\frac{ x }{ y} })=(0+e^{\frac{ x }{ y }}(x \frac{ ∂ }{ ∂y }y^{-1}+y^{-1}\frac{ ∂ }{ ∂y }x))\]

Answer 17

it says differentiate all u need is implicit differentiation

Answer 18

"x" is a constant. and whats the answer when u take the deriv of a constant?

Answer 19

wait, digesting your answer O_O

Answer 20

soo, i need to bring the whole power on exponent to infront exponent ?

Answer 21

In order for the the equation to be exact
\[M_y=N_x\] must hold true.

Answer 22

normally we will bring the constant ... oh, implicit ?

Answer 23

where \[M_y , N_x\] is the partial derivatives

Answer 24

so, i need to do implicit for the power, and bring it infront ?

Answer 25

You're not doing implicit differentiation, you're taking partial derivatives.

Answer 26

._.

Answer 27

have u done partial derivative

Answer 28

err, im not familiar with that term ... ._.

Answer 29

ok so u need implicit differentiation then

Answer 30

probably because my lecturer used to say it in our language, but ...

Answer 31

that's what i said above

Answer 32

ohh

Answer 33

let @abb0t solve it i am sure he can do it

Answer 34

referring to the picture given, so, partial is product rule ? D:

Answer 35

but that is a long solution

Answer 36

no forget partial derivatives

Answer 37

it is another interesting topic of calculus

Answer 38

O.o oh i see

Answer 39

NO partial is not product rule
i don't know how your instructor is supposed to start Exac equation without discussing the partial derivatives . wait let me write little bit about it below.

Answer 40

well in ur implicit differentiation u need the product rule

Answer 41

i don't see how u could do implicit derivs with dx already there.

Answer 42

it is not ur fault @abb0t, it is the instructor way of writing questions

Answer 43

i have seen many questions written the wrong way by teachers...

Answer 44

\[(1+e^{\frac{ x }{ y }})+e^\frac{ x }{ y }(1+asd)\frac{ dy }{ dx }=0 \]

Answer 45

this is a tricky question

Answer 46

._.

Answer 47

i will solve it in details

Answer 48

first of all let's break this down a bit ok

Answer 49

Ok, first check if they are exact. idk wat the partial derivatives are, but it looks like they are exact.

Answer 50

wait, can i ask 1 question, base on what @sami-21 uploaded before

Answer 51

\[\frac{d}{dx}e^{\frac{x}{y}}\]

Answer 52

nevermind, go on :D

Answer 53

this part u need the chain rule...

Answer 54

@sami-21 is doing it correctly.

Answer 55

wow, ok, i can understand it ... but ... where do we apply it ? O.o

Answer 56

i guess my lecturer taught us a little bout this, but too brief ._. the way @sami-21 explain much more easy to understand

Answer 57

let \[u = \frac{x}{y} \space \therefore \frac{d}{dx}e^u \frac{du}{dx} \space and \space \frac{d}{du}e^u\]

Answer 58

till now you are using it just for Exact equations . in your susequent courses of Mathematics you study this in detail.

Answer 59

Yes, if you did implicit differentiation, you would get y'' and x'' and that would not work for exact equations.

Answer 60

Anyways, I will help you find the solution but you should review partial derivatives.

Answer 61

still wondering where do i have to apply this partial derivative ... ._.

Answer 62

i don't think the guy has ever done partial derivative

Answer 63

\[\int\limits Mdx = \int\limits (1+e^{\frac{ x }{ y }})dx\]

Answer 64

i quite understand the concept ...
and ... so, its e^{x/y} ?

Answer 65

which concept implicit or partial?

Answer 66

partial

Answer 67

\[\int\limits (1+e^{\frac{ x }{ y} })dx = x + ye^{\frac{ x }{ y }}+h(y)\]

Answer 68

with partial derivative the solution is much easier...

Answer 69

Since u are working with TWO variables here and talking about partial derivs with respect to x, this means that any term that contained only constants or y’s would have differentiated away to zero, therefore u need to acknowledge that fact by adding on a function of y instead of the standard c.

Answer 70

Now you need to solve for h(y) by using the same method \[\frac{ ∂ }{ ∂y }( x+ye^{\frac{ x }{ y }}+h(y))=N\]

Answer 71

partial derivatives again, this time, x is ur constant instead of y. so you have: \[0+(y \frac{ ∂ }{ ∂y }e^{\frac{ x }{ y} }+e^\frac{ x }{ y}) = N\]

Answer 72

actually, i've been thinking bout this ...
\[e ^{x/y}\]

Answer 73

here is full detail solution.

Answer 74

oh wait, ur just asked to show it's exact. Lol. nvm hahaha.

Answer 75

oh, u hit the answer @sami-21, but im trying to understand it.

Answer 76

ignore the long question, i only concern bout \[e ^{x/y}\] part

Answer 77

is this the part where im suppose to apply partial ?

Answer 78

when taking partial derivative one variable is treated as a constant.

Answer 79

therefore, you would use chain rule and then find the derivative with x or y as a constant. depending on what partial fraction ur being asked to find.

Answer 80

base on this http://assets.openstudy.com/updates/attachments/51315608e4b0f0611bc02418-sami-21-1362193507528-par.png

Answer 81

i didn't understand what's the purpose of seperating those, okay, lets back to this, e^(x/y) is that mean i've to seperate it like this :-

(e^x)(e^1/y) ?

Answer 82

but it will not change anything though -_-"

Answer 83

you need to know how to differentiate e^(x/y) ?

Answer 84

yeahh, exactly !

Answer 85

oh, finally understand this :D