Results 1 to 8 of 8

Math Help - Derivative qus.

  1. #1
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1

    Derivative qus.

    Hey guys,

    Finding hard to solve these derivative:

    1)  q = 2e^{-t/2}cos2t (don't know how to deal with the index -t/2)

    2)  x = e^{2t}t^3(2-t)^4 (this seem confusing as don't know what rule to apply)

    3)  e^{x+y} + sin3x = 0 (not to sure about index agian)

    I hope someone can help. Thanks

    dadon
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dadon
    1)  q = 2e^{-t/2}\cos(2t) (don't know how to deal with the index -t/2)
    Start with the product rule:

    <br />
\frac{dq}{dt}=2 \left[\left(\frac{d}{dt}e^{-t/2}\right)\cos(2t)+e^{-t/2}\left(\frac{d}{dt} \cos(2t)\right)\right]<br />

    so using the chain rule to do the derivatives:

    <br />
\frac{dq}{dt}=2 \left[-\frac{1}{2}e^{-t/2}\cos(2t)-2e^{-t/2}\sin(2t)\right]<br />

    simplifying:

    <br />
\frac{dq}{dt}=-e^{-t/2}[\cos(2t)+4\sin(2t)]<br />

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dadon
    Hey guys,

    Finding hard to solve these derivative:

    1)  q = 2e^{-t/2}cos2t (don't know how to deal with the index -t/2)

    2)  x = e^{2t}t^3(2-t)^4 (this seem confusing as don't know what rule to apply)

    3)  e^{x+y} + sin3x = 0 (not to sure about index agian)

    I hope someone can help. Thanks

    dadon
    What you seem to be having a problem with in 1) and 3) is:

    <br />
\frac{d}{dx}e^{f(x)}<br />

    By the chain rule this is:

    <br />
\frac{d}{dx}e^{f(x)}=f'(x)\ \frac{d}{dx}e^{y}\left\vert_{f(x)}\frac{}{}=f'(x)\ e^{f(x)}<br />

    RonL
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1

    Post re:

    Thanks for the quick response and the general rule!

    So for 2) is it product and chain together?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dadon
    Thanks for the quick response and the general rule!

    So for 2) is it product and chain together?
    Yes, though you will have the product:

    <br />
x = \{e^{2t}\}\ \{t^3(2-t)^4\}<br />

    so you may need to use the product rule again for the derivative of the
    second term on the RHS.

    RonL
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by dadon

    3)  e^{x+y} + sin3x = 0 (not to sure about index agian)
    This is an example of implicit differentiation (and the chain rule):

    <br />
\frac{d}{dx} e^{x+y}+\frac{d}{dx}\sin(3x)=0<br />
,

    so:

    <br />
e^{x+y}\ \frac{dy}{dx} + 3\cos(3x)=0\ \ \ \dots(1)<br />

    but from the original equation we know that:

    <br />
e^{x+y}=-\sin(3x)<br />
,

    so substituting this into equation (1) gives:

    <br />
-\sin(3x)\frac{dy}{dx}+3\cos(3x)=0<br />

    or rearranging:

    <br />
\frac{dy}{dx}=3\frac{\cos(3x)}{\sin(3x)}=3\cot(3x)<br />

    RonL
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1
    Hi Thanks again!

    Both methods should give the same answer?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Apr 2006
    Posts
    201
    Awards
    1
    The implicit way is a much neater, when applying the product and chain rule it gets abit messy that's what I have been trying!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. contuous weak derivative $\Rightarrow$ classic derivative ?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 22nd 2011, 03:37 AM
  2. Replies: 0
    Last Post: January 24th 2011, 12:40 PM
  3. Replies: 2
    Last Post: November 6th 2009, 03:51 PM
  4. Replies: 1
    Last Post: January 7th 2009, 02:59 PM
  5. Fréchet derivative and Gâteaux derivative
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 23rd 2008, 05:40 PM

Search Tags


/mathhelpforum @mathhelpforum