Calculus: Finding First Derivatives for Functions - Homework Help

[DTskkaii]

Hi, I was hoping someone could help me out with my homework set.
I have done a lot of the questions, and it would help if someone could tell me if I have done them correctly. Thanks! :)

Q1: Find first derivatives for the following functions
(a)g(s,t)=sin(st^3)
g_t=δg/δt=δ/δt*(sin(st^3))=t^3cos(st^3)
g_s=δg/δs=δ/δs*(sin(st^3))=3st^2cos(st^3)
Just wrote that out full for working, will shorten now
(b) f(x,y)=x(y^3)+(2x^4)y
f_x=(y^3)+8(x^3)y
f_y=3x(y^2)+2(x^4)
(c) g(r,x,z)=rsin(zx)
g_r=sin(zx)
g_x=rzcos(zx)
g_z=rxcos(zx)
(d) e_(X1,X2...Xn)=sqrt(X1^2+X2^2...Xn^2)
only write down one generalised partial derivative with respect to Xi
I am not sure how to approach this one, help would be great
I also am not entirely sure if I am somehow meant to put the two partial derivatives back together to get the complete first derivative?

Q2: y(x,t)=Asin(kx-wt) where w=(pi/2), k=pi, A=5
(a) find rate of change of y wrt to t at x=1, t=1
δy/δt=-Awcos(kx-wt)
=-(5/2)pi*cos(pi/2)
=-7.851
(b) find rate of change of y wrt to x at x=(1/2), t=1
δy/δx=-Akcos(kx-wt)
=-5pi*cos0
=15.708

If anyone could tell me if I am on the right track with these questions, and help out with Q1 (d), I would be super appreciative.

 

[Ray Vickson]

Hi, I was hoping someone could help me out with my homework set.
I have done a lot of the questions, and it would help if someone could tell me if I have done them correctly. Thanks! :)

Q1: Find first derivatives for the following functions
(a)g(s,t)=sin(st^3)
g_t=δg/δt=δ/δt*(sin(st^3))=t^3cos(st^3)
g_s=δg/δs=δ/δs*(sin(st^3))=3st^2cos(st^3)
Just wrote that out full for working, will shorten now
(b) f(x,y)=x(y^3)+(2x^4)y
f_x=(y^3)+8(x^3)y
f_y=3x(y^2)+2(x^4)
(c) g(r,x,z)=rsin(zx)
g_r=sin(zx)
g_x=rzcos(zx)
g_z=rxcos(zx)
(d) e_(X1,X2...Xn)=sqrt(X1^2+X2^2...Xn^2)
only write down one generalised partial derivative with respect to Xi
I am not sure how to approach this one, help would be great
I also am not entirely sure if I am somehow meant to put the two partial derivatives back together to get the complete first derivative?

Q2: y(x,t)=Asin(kx-wt) where w=(pi/2), k=pi, A=5
(a) find rate of change of y wrt to t at x=1, t=1
δy/δt=-Awcos(kx-wt)
=-(5/2)pi*cos(pi/2)
=-7.851
(b) find rate of change of y wrt to x at x=(1/2), t=1
δy/δx=-Akcos(kx-wt)
=-5pi*cos0
=15.708

If anyone could tell me if I am on the right track with these questions, and help out with Q1 (d), I would be super appreciative.

Your computation of g_t in (a) is incorrect; you need to use the Chain Rule to get a proper computation. You should have gotten $g_t = 3 s t^2 \cos(st^3).$ I did not check the others, so there may or may not be additional errors.

RGV

 

[micromass]

DTskkaii,

It may be nice to use LaTeX in your post. That would make it so much more readable and easy for us!

https://www.physicsforums.com/showthread.php?t=546968

 

[DryRun]

DTskkaii,

It may be nice to use LaTeX in your post. That would make it so much more readable and easy for us!

https://www.physicsforums.com/showthread.php?t=546968

I'm glad you posted this link, as i just made it into my signature. Hopefully, it will help spread the message.