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Math Help - logarithmic differentiaion

  1. #1
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    logarithmic differentiaion

    Hey guys my teacher is confusing the hell out of me with this one problem we went over in class.So here it is.

    f(x) = (2x + 1)^10 * (3x +1)^20 / sinx sqrt(cosx)

    Please help the test is on friday.
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  2. #2
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    logarithmic differentiaion

    Hey guys my teacher is confusing the hell out of me with this one problem we went over in class.So here it is.

    f(x) = (2x + 1)^10 * (3x +1)^20 / sinx sqrt(cosx)

    Please help the test is on friday.
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  3. #3
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by uniquereason81 View Post
    Hey guys my teacher is confusing the hell out of me with this one problem we went over in class.So here it is.

    f(x) = (2x + 1)^10 * (3x +1)^20 / sinx sqrt(cosx)

    Please help the test is on friday.
    Is everything in the numerator divided by sinx(cosx)^.5?
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  4. #4
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    yes but its sinx * square root(cosx)
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  5. #5
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by uniquereason81 View Post
    Hey guys my teacher is confusing the hell out of me with this one problem we went over in class.So here it is.

    f(x) = (2x + 1)^10 * (3x +1)^20 / sinx sqrt(cosx)

    Please help the test is on friday.
    So it looks like this

     f(x) = \frac{(2x + 1)^{10}(3x +1)^{20}}{\sin{x}\sqrt{\cos{x}}}
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  6. #6
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    you got it
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  7. #7
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by 11rdc11 View Post
    So it looks like this

     f(x) = \frac{(2x + 1)^{10}(3x +1)^{20}}{\sin{x}\sqrt{\cos{x}}}
    Ok so first thing we do is take ln of both sides

     \ln{y} = \ln{\frac{(2x + 1)^{10}(3x +1)^{20}}{\sin{x}\sqrt{\cos{x}}}}

    Next use the properties of ln to simplify the right side

     \ln{y} = \ln{(2x + 1)^{10}(3x+1)^{20}} -\ln{\sin{x}\sqrt{\cos{x}}}

    \ln{y} = \ln{(2x+1)^{10}} + \ln{(3x+1)^{20}}-[\ln{\sin{x}} + \ln{\cos^{\frac{1}{2}}{x}}]

    \ln{y} = 10\ln{(2x+1)} + 20\ln{(3x+1)}-\ln{\sin{x}} - \frac{\ln{\cos{x}}}{2}

    Can you finish up from here?
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  8. #8
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    do u do the derivative of y on both sides and then move y to the other side
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  9. #9
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by uniquereason81 View Post
    Hey guys my teacher is confusing the hell out of me with this one problem we went over in class.So here it is.

    f(x) = (2x + 1)^10 * (3x +1)^20 / sinx sqrt(cosx)

    Please help the test is on friday.
    Please don't double post. I'm working it for you on your other post.
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  10. #10
    Super Member 11rdc11's Avatar
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    Yep

    \ln{y} becomes \frac{y'}{y}

    so y' = y(the derivative of the right side)

    y= \frac{(2x + 1)^{10}(3x +1)^{20}}{\sin{x}\sqrt{\cos{x}}}

    Do you need me to workout the derivative?
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  11. #11
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    Hello, uniquereason81!

    This is a messy one.
    I must assume you know the basics of Logarithmic Differentiation.


    y \;=\;\frac{(2x + 1)^{10}(3x +1)^{20}}{\sin x \sqrt{\cos x}}
    Take logs and expand as much as possible . . .

    \ln(y) \;=\;\ln\left[\frac{(2x+1)^{10}(3x+1)^{20}}{\sin x \sqrt{\cos x}} \right]

    . . . . = \;\ln\bigg[(2x+1)^{10}(3x+1)^{20}\bigg] - \ln\bigg[\sin x (\cos x)^{\frac{1}{2}}\bigg]

    . . . . = \;\ln(2x+1)^{10} + \ln(3x+1)^{20} - \bigg[\ln(\sin x) + \ln(\cos x)^{\frac{1}{2}}\bigg]

    . . . . = \;10\ln(2x+1) + 20\ln(3x+1) - \ln(\sin x) - \frac{1}{2}\ln(\cos x)


    Differentiate implicitly:

    . . \frac{1}{y}\cdot\frac{dy}{dx} \;=\;10\cdot\frac{2}{2x+1} + 20\cdot\frac{3}{3x+1} - \frac{\cos x}{\sin x} - \frac{1}{2}\cdot\frac{\text{-}\sin x}{\cos x}

    . . \frac{1}{y}\cdot\frac{dy}{dx} \;=\;\frac{20}{2x+1} + \frac{60}{3x+1} - \cot x + \frac{1}{2}\tan x


    Multiply by y\!:\quad\frac{dy}{dx} \;=\;y\left[\frac{20}{2x+1} + \frac{60}{3x+1} - \cot z + \frac{1}{2}\tan x\right]


    Therefore: . \frac{dy}{dx}\;=\;\frac{(2x+1)^{10}(3x+1)^{20}}{\s  in x\sqrt{\cos x}}\left[\frac{20}{2x+1} + \frac{60}{3x+1} - \cot x + \frac{1}{2}\tan x\right]

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  12. #12
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    yes please cause i don't think he went that far cause we ran out of time
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  13. #13
    Super Member 11rdc11's Avatar
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    http://www.mathhelpforum.com/math-he...rentiaion.html

    Soroban finished it for you on your other post. If you have have any more questions feel free to ask.
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  14. #14
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    Quote Originally Posted by uniquereason81 View Post
    Hey guys my teacher is confusing the hell out of me with this one problem we went over in class.So here it is.

    f(x) = (2x + 1)^10 * (3x +1)^20 / sinx sqrt(cosx)

    Please help the test is on friday.
    Don't double post. It wastes everyone's time. The evidence for you wasting people's time is here: http://www.mathhelpforum.com/math-he...tml#post203121
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