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Calculus; Derivatives

  1. May 21, 2012 #1
    Hi, I was hoping someone could help me out with my homework set.
    I have done alot 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 im on the right track with these questions, and help out with Q1 (d), I would be super appreciative.
     
    Last edited: May 21, 2012
  2. jcsd
  3. May 21, 2012 #2

    Ray Vickson

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    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 [itex]g_t = 3 s t^2 \cos(st^3).[/itex] I did not check the others, so there may or may not be additional errors.

    RGV
     
  4. May 21, 2012 #3

    micromass

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  5. May 22, 2012 #4

    sharks

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