Question

An object moves along the path x^2 - y^2 = 5. At the point (-3, 2), dx/dt=-4. Find dy/dt at that same point.

Answer 1

from the equation you can differentiate x on both sides and get dy/dx=f(x,y), which is actually a number is x and y are known

Answer 2

now, dy/dx=dy/dt*dt/dx, since dt/dx=1/(dx/dt), there is only one unkown which is dy/dt, solve it and you are done

Answer 3

Umm I'm still confused..

Answer 4

\[2x-2yy'=0\] correct?

Answer 5

Yes :)

Answer 6

yes that's correct

Answer 7

now solve for dy/dt

Answer 8

but you missed dx/dt

Answer 9

\[y'=\frac{ x }{ y }\]

Answer 10

if x=-3,y=2, y'=-3/2

Answer 11

x and y are all depend on t

Answer 12

so dy/dt*dt/dx=-3/2

Answer 13

since dx/dt=-4,dt/dx=1/-4

Answer 14

\(x^2-y^2=5\)
\(2xx'-2yy'=0\)

Answer 15

dy/dt*-1/4=-3/2, dy/dt=6

Answer 16

we have diff notation, my y' is dy/dx, not dy/dt

Answer 17

solving for y'
\(y'=xx'/y\)

Answer 18

now substitute

Answer 19

Ohhhhh! Now I understand how to do dy/dt! :)

Answer 20

and we arrive at same answer, but diff method

Answer 21

why are looking dy/dx?
the question is asked to find dy/dt

Answer 22

That makes so much more sense thank you guys! :)

Answer 23

eh i just realized @Breee615 is the poster haha
thought it was you

Answer 24

because I can use chain rule

Answer 25

yes you can! so you diff with respect to x first the implicit equation
is that what you did?

Answer 26

using x' is ambiguous, but works as long as you know it means dx/dt
But being more explicit , and using implicit differentiation with respect to t:
\[ \frac{d}{dt} \left(x^2 - y^2 = 5\right) \\2x \frac{dx}{dt} - 2y \frac{dy}{dt}= 0
\]
now fill in the known values, and solve for dy/dt

Answer 27

yeah true, i wanted to write that way but kind lazy to use too much latex hahah

Answer 28

yes, that exactly what I did, then I find out the dy/dx= some number by pluging in x and y, finally the dx/dt comes in during chain rule

Answer 29

but certainly i agree with you

Answer 30

So wait my answer is 6 right?? If it isn't then I must not understand how to do it :o

Answer 31

Yes it is

Answer 32

oh makes sense :)

Answer 33

you should have used dy/dx instead of y' hahah

Answer 34

Okay good, thank you :)

Answer 35

that's chain rule still

Answer 36

so how did you get -12

Answer 37

the answer 6 for sure haha

Answer 38

the first step is correct old man :)
what the next step

Answer 39

oh that's the parametric thingy

Answer 40

oh great job!
you just did what @coazeyuan did
just you nailed it down more

Answer 41

you used the parametric derivative dy/dx

Answer 42

you can use several variable calc but the poster should understand it though

Answer 43

Do you guys mind helping me with another problem quick? :)

Answer 44

post in a diff post

Answer 45

Okay! :)

Answer 46

and tag