Question

for the equation given below, y/x+8y=x^9-4 evaluate dy/dx at point (1,-3/25)

Answer 1

Troubles differentiating I suppose?

Answer 2

\[\frac{y}{x}+8y=x^9-4 \text{ is correct interpretation ?} \]

Answer 3

sorry i meant (y)/(x+8y)=x^9-4

Answer 4

8y under the y as well

Answer 5

\[\frac{y}{x+8y}=x^9-4\]
I bet you have no problem with the right hand side

Answer 6

The left hand side will require us to know quotient rule

Answer 7

\[\frac{d}{dx}(\frac{y}{x+8y})=\frac{(y)'(x+8y)-y(x+8y)'}{(x+8y)^2}\]

Answer 8

This is only leaves you with differentiating y and (x+8y)

Answer 9

\[\frac{d}{dx}y=\frac{dy}{dx} \text{ (note: which people also call this) } =y'\]

Answer 10

Try finding
\[\frac{d}{dx}(x+8y)=?\]

Answer 11

ok i got,
[1+8 d/dx (y(x))\]

Answer 12

ok so we have 1+8y' coolness

Answer 13

ok so let's put all of that in...

Answer 14

\[\frac{y'(x+8y)-y(1+8y')}{(x+8y)^2}=9x^8\]

Answer 15

Now we need to solve for y'...

Answer 16

first I would multiply both side by that ickyness in the bottom on the left side

Answer 17

Then after you have done
I would start using distributive property where needed on left.
Then gather my terms that y' and put everything else on the other side

Answer 18

i dont get what you mean by multiply both sides

Answer 19

multiply both sides by what?

Answer 20

by the ickyness on the bottom on the left

Answer 21

There is a fraction on the left
a fraction has a top and a bottom

Answer 22

\[\cancel{(x+8y)^2} \cdot \frac{y'(x+8y)-y(1+8y')}{\cancel{(x+8y)^2}}=9x^8(x+8y)^2\]

Answer 23

I multiplied both sides by the ickyness that was on the bottom of the left

Answer 24

I'm doing this because I'm trying to isolate y'

Answer 25

That is the side with y'

Answer 26

To get to y' I have to undo that division first

Answer 27

oh ok..

Answer 28

got you

Answer 29

now do i distribute on the left?

Answer 30

Yep
If you can actually skip that step if you are really good with algebra and sorta like basically do it in your head
but yeah

Answer 31

You want to gather your terms that have y' together

Answer 32

xy'+8yy'-7+8yy'?

Answer 33

is that correct for the left side?

Answer 34

\[xy'+8yy'-y-8yy'\]

Answer 35

yah i meant y instead of 7 lol

Answer 36

and a - in front of the 8yy'

Answer 37

but anyways you should see something cancel there

Answer 38

before we move on

Answer 39

so xy'-y

Answer 40

\[xy'-y=9x^8(x+8y)^2\]

Answer 41

Do you think you can finish solving for y' now?

Answer 42

You only need 2 more steps

Answer 43

add y and divide by x correct?

Answer 44

yep

Answer 45

\[y'=\frac{9x^8(x+8y)^2+y}{x}\]

Answer 46

yep now i will plug in the x and y values

Answer 47

and see if its right

Answer 48

yeah

Answer 49

ok one sec

Answer 50

-1.056

Answer 51

and it is correct

Answer 52

Well I will have to check again
I got -.1056
maybe i made mistake then

Answer 53

thank you freckles! can you help me with another question

Answer 54

no it is right, i submitted it and it came up correct

Answer 55

oh I thought you put in -1.056

Answer 56

oh no .-1056

Answer 57

ok cool

Answer 58

suppose that (g(x))^2+6x=x^2g(x)+17 and g(2) =5 find g'(2)