Partial F/ Partial T F= (x,y) x and y = functs of s and t Only step 1 needed

[Yezman]

If f(x,y) = Sqrt[x^4 + y] and x = s^2 + t^2 + 1 and y = t^2 +t*Cos(2s)

How do i find

Partial f
--------
Partial tI made the tree diagram how f depends on x and y, which both depend on s and t... so on my test I said it was

(Partials of all of these)f/x*x/s*x/t + f/y*y/s*y/tLooking back now I wonder why I put the s's in there, seems like I don't need them. (this was from a recent test and we get to redo 1 problem). My new idea is

f/x*x/t + f/x*x/t ? That seems to simple though

Can someone here point in the right direction?

Thanks

 

[SammyS]

Think of this as:

f(x(s),y(t)) = \sqrt{(x(s,t))^4 + y(s,t)}\,,\text{ where } x(s,t) = s^2 + t^2 + 1\text{ and }y(s,t) = t^2 +t\,\,\cos(2s)

 

[LCKurtz]

If f(x,y) = Sqrt[x^4 + y] and x = s^2 + t^2 + 1 and y = t^2 +t*Cos(2s)

How do i find

Partial f
--------
Partial t


I made the tree diagram how f depends on x and y, which both depend on s and t... so on my test I said it was

(Partials of all of these)


f/x*x/s*x/t + f/y*y/s*y/t


Looking back now I wonder why I put the s's in there, seems like I don't need them. (this was from a recent test and we get to redo 1 problem). My new idea is

f/x*x/t + f/x*x/t ? That seems to simple though

Can someone here point in the right direction?

Thanks

If what you mean by that last equation is

f'(t) = (f_x)*x'(t) + (f_y)*y'(t)

that is the correct formula to work the problem

 

[Yezman]

Is that an f prime? I am looking for ∂f/∂t

I re-read the section (stewart Calc)

Wouldn't it just be


∂f/∂x* ∂x/∂t + ∂f/∂y* ∂y/∂t ?

 

[LCKurtz]

Is that an f prime? I am looking for ∂f/∂t

I re-read the section (stewart Calc)

Wouldn't it just be


∂f/∂x* ∂x/∂t + ∂f/∂y* ∂y/∂t ?

Yes, exactly. I didn't notice that x and y were also functions of s in addition to t.

 

[Yezman]

Ok... Idk what i was thinking when i had to multiply three things at a time (lack of sleep probably)Just to check... if you had

(I made this up so something might be weird)
∂f/∂r = ?

f(x,y,z) & x =(s,t,r) & y =(s,t,r) & z =(s,t,r)

∂f/∂r = ∂f/∂x*∂x/∂r + ∂f/∂y*∂y/∂r + ∂f/∂z*∂z/∂r ?

 

[LCKurtz]

Ok... Idk what i was thinking when i had to multiply three things at a time (lack of sleep probably)


Just to check... if you had

(I made this up so something might be weird)
∂f/∂r = ?

f(x,y,z) & x =F(s,t,r) & y =G(s,t,r) & z =H(s,t,r)

∂f/∂r = ∂f/∂x*∂x/∂r + ∂f/∂y*∂y/∂r + ∂f/∂z*∂z/∂r ?

Yes, you have the idea. I inserted functions for you.