Question

find dy/dx by implicit differentiation y^4+xy^2=5+x^3y

Answer 1

Implicit differentiation just means that you need to find a dy/dx from the equation, which is also seen as y'. Knowing this, if we differentiate the enter problem we should be able to find dy/dx afterwards by using algebra.

Answer 2

Is that:\[y^4+xy^2=5+x^3y\]or\[y^4+xy^2=5+x^{3y}\]

Answer 3

first one

Answer 4

Since you don't know y, really when you're differentiating it you use the chain rule. That's the only thing you do in implicit differentiation.

Answer 5

can you walk me through that?

Answer 6

When you finding the derivative to respect to y, and you differentiate two sides of an equation, each x would differentiate normally. Ex ) x^2 = 2x. The tricky part is finding what the y's turn into.

Answer 7

So do you get what @Coda is saying?
\[\partial (x^2y^2)=2xy^2\partial x+x^22y\partial y\]

Answer 8

not really

Answer 9

So when you see something like 5y^2=3x^2 all you are really doing is saying

(10y)dy/dx=6x*1

See on the left side you did the derivative as normal, the outside stuff, then multiplied by the derivative of the inside. The derivative of y is just dy/dx!

Answer 10

Or to back up one step with @Kainui
\[5y^2=3x^2\]\[10y \ dy=6x\ dx\]

Answer 11

Don't try to reconcile the dy/dx until the end. First just take the derivatives of both sides.

Answer 12

Or of each term as if each term were a separate derivative.

Answer 13

So let's start with the first term y^4. What is the derivative of y^4?

Answer 14

@Coda I didn't say anything incorrect, I just gave an example.

Also, remember that you're using the product rule and the chain rule at the same time, so be careful! The chain rule just means one of the products times the derivative of one plus the derivative of the other times the other kept constant.

Answer 15

Sorry @Kainui, I forgot to add my commas or they did not register.

Answer 16

@jeanniebean what dont you get here?

Answer 17

umm i just need an example with this problem with the two people trying to help me i got confused and just have alot of stuff written down can we start at the beginning and go step by step?

Answer 18

\[y^4 + xy^2 = 5 + x^3 y\]
what is the derivative of y^4?

Answer 19

4y^3

Answer 20

That's almost right, but we assume for the moment that y is a function of x, so, after you're done differentiating y as you would x, you affix a dy/dx :)

Answer 21

right but dont forget y'

Answer 22

so 4dy/dx^3?

Answer 23

no.. \(4y^3 y'\)

Answer 24

oh ok

Answer 25

No, just differentiate it as you would x,which is what you did...
4y^3 and then when you're done, multiply the whole thing with dy/dx, or y'

Answer 26

next what is the derivative of xy^2? use product rule here

Answer 27

ok

Answer 28

x(2y*y')+1(y^2)

Answer 29

That's right :) I take it you can do the rest?

Answer 30

what do i do after i get 4y^3(y')+x(2y*y')+1(y^2)= 0+x^3(y')=3x^2(y)

Answer 31

put y' on one side and remaining on the other side of equal to..

Answer 32

how do i do that with it being so many?

Answer 33

what are the terms with y'?

Answer 34

4y^3(y')+x(2y*y')=x^3(y')

Answer 35

i guess u left out y^2
take common from where u cn...result according to me is \[-y ^{2}/(4y ^{3}+2xy-x ^{3)}\]

Answer 36

what is that the result for?

Answer 37

that is y' or dy/dx

Answer 38

oh so like thats the answer?

Answer 39

thats all you have to do?

Answer 40

according to me..yes!

Answer 41

oh ok well can you help me with integrals?

Answer 42

i cn try integrals...

Answer 43

ok : \[\int\limits_{?}^{?} (2x ^{2}+ 5\sec ^{2}x)dx=\]

Answer 44

there is nothing where the question marks are

Answer 45

i thnk...answer is \[x ^{4}/2 + 5 tanx\]

Answer 46

can you write out how you got to that?

Answer 47

yeah sure..integral of any nx^y is (nx^(y+1))/y+1 and integral of sec^2 x is tan x....
so...integral of 2x^3 should hv been \[(2x ^{3+1})/(3+1)\]

Answer 48

oh and then you just get the answer?

Answer 49

yup!

Answer 50

i thnk...!

Answer 51

good enough for me if i give you another one can you try it and just write out the steps that you took to get there too?

Answer 52

sure!

Answer 53

\[\int\limits_{1}^{2}(1-2x ^{2}+x ^{5})/x ^{5} dx=\] it is the part in parentheses over the x^5 sorry i dont know how to make a fraction

Answer 54

does that make sense?

Answer 55

umm...(1-2x^3+x^5)/x^5 is the fraction??

Answer 56

yeah like the part in parentheses over the x^5

Answer 57

ohk...w8!

Answer 58

ok

Answer 59

i htnk i ll do it wrong...i m yet to do my integration classes in school...

Answer 60

oh ok