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lovesit2x

  • one year ago

Differentiate y = e^(4x) cos x with respect to x dy/dx = ?

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  1. zepdrix
    • one year ago
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    Hmm looks like we'll have to apply the product rule :)

  2. Xavier
    • one year ago
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    You have a product so use the product rule

  3. zepdrix
    • one year ago
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    \[\large y=e^{4x}\cos x\]\[\large y'=\color{royalblue}{\left(e^{4x}\right)'}\cos x+e^{4x}\color{royalblue}{\left(\cos x\right)'}\] Understand the setup for Product Rule? We have to take the derivative of the blue terms.

  4. lovesit2x
    • one year ago
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    Yes

  5. zepdrix
    • one year ago
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    So what's the derivative of cos x? :)

  6. lovesit2x
    • one year ago
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    sin x ?

  7. lovesit2x
    • one year ago
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    wait -sinx

  8. zepdrix
    • one year ago
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    Yes good c:\[\large y'=\color{royalblue}{\left(e^{4x}\right)'}\cos x+e^{4x}\left(-\sin x\right)\]

  9. zepdrix
    • one year ago
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    Do you remember the derivative of \(\huge e^x\) ?

  10. lovesit2x
    • one year ago
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    Nope :)

  11. zepdrix
    • one year ago
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    Oooo tsk tsk! That's a fun easy one that you'll want to remember! c: \[\huge \left(e^x\right)'=e^x\] It gives us the same thing back. This same thing will happen in our problem here, except we'll have an extra step. Since our exponent is more than just \(\large x\), we have to apply the chain rule. Multiply the result by the derivative of the exponent. \[\huge \left(e^{4x}\right)'=e^{4x}\left(4x\right)'\]

  12. zepdrix
    • one year ago
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    Hmm so what is the derivative of that exponent. The derivative of \(\large 4x\).... hmmmm

  13. lovesit2x
    • one year ago
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    4x

  14. zepdrix
    • one year ago
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    The derivative of 4x is 4x? Hmm no that's not going to work :c

  15. lovesit2x
    • one year ago
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    4

  16. lovesit2x
    • one year ago
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    lol

  17. zepdrix
    • one year ago
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    Yessss good :) Giving us, \(\huge e^{4x}(4)\) Which gives us an answer of,\[\huge y'=4e^{4x}\cos x+e^{4x}\left(-\sin x\right)\]

  18. lovesit2x
    • one year ago
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    WOW...you just made math fun. lol Thanks

  19. zepdrix
    • one year ago
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    lol :3 np

  20. lovesit2x
    • one year ago
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    wait...its with respect to x

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