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Math Help - Checking work (derivatives)

  1. #1
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    Checking work (derivatives)

    I have the binomial

    (t^2+6)(1+t^2)

    working out the problem I have ...

    t^4+7t^2+6

    y'= y'=4t^3+14t

    = 2t(2t^2+7)

    Is this correct?

    Using the product rule, I get the same thing

    (2t)(1+t^2)+(2t)(t^2+6)

    = 2t+2t^3+2t^3+12t = 4t^3+14t

    = 2t(2t^2+7)

    Sorry, I got a multiple choice assignment and I want to make extra sure as the answers are all very close or changing a sign here or something subtle
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  2. #2
    Super Member wingless's Avatar
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    Perfect!
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  3. #3
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    Hello, XIII13Thirteen!

    I have the function: . y \;=\;(t^2+6)(1+t^2)

    Working out the problem I have:

    . . y \;=\;t^4+7t^2+6\quad\Rightarrow\quad y'\:=\:4t^3+14t  \:=\:2t(2t^2+7)

    Is this correct? . . . . . Yes!


    Using the product rule, I get the same thing:

    y' \;=\;(2t)(1+t^2)+(2t)(t^2+6) \:=\:2t+2t^3+2t^3+12t

    . . = \:4t^3+14t \:=\:2t(2t^2+7) . . . . . (Yes!)

    That is an excellent way to check your work!


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    I have told my students that they can easily create practice problems
    . . for the Product Rule and Quotient Rule.


    We know that: . y \;=\;x^9\quad\Rightarrow\quad y' \;=\;9x^8


    \text{Now use the Product Rule on: }\;y \;=\;\overbrace{(x^4)}^{f(x)}\overbrace{(x^5)}^{g(  x)}

    \text{We have: }\;y'\;=\;\overbrace{(x^4)}^{f(x)}\overbrace{(5x^4  )}^{g'(x)} + \overbrace{(4x^3)}^{f'(x)}\overbrace{(x^5)}^{g(x)} \;=\;5x^8 + 4x^8 \;=\;9x^8\qquad\hdots\;see?


    \text{Now use the Quotient Rule on: }\;y \;=\;\frac{\overbrace{x^{12}}^{f(x)}}{\underbrace{  x^3}_{g(x)}}

    \text{We have: }\;y' \;=\;\frac{\overbrace{(x^3)}^{g(x)}\overbrace{(12x  ^{11}}^{f'(x)}) - \overbrace{(x^{12})}^{f(x)}\overbrace{(3x^2)}^{g'(  x)}}{\underbrace{(x^3)^2}_{[g(x)]^2}} \;=\;\frac{12x^{14}-3x^{14}}{x^6} \;=\;\frac{9x^{14}}{x^6} \;=\;9x^8

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  4. #4
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    @ Soroban: Thanks for giving me an extended view of the concepts as well. I think the more I drill this stuff into myself the more confidant I'll feel about it
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