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Math Help - Find dy/dx

  1. #1
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    Find dy/dx

    Question : Find \frac{dy}{dx}

     y = \frac{(x+2)^3 (3x+5)^{-4} sinx}{(2x+2)^2}
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  2. #2
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    differentiate :

    \ln y = 3 \ln (x+2) - 4\ln(3x+5) + \ln(\sin x) - 2\ln (2x+2)
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  3. #3
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    hmm thats a neat way of interpreting a derivative thanks for that gives me a new perspective on computing really long derivatives.
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  4. #4
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    I am stuck here

    Quote Originally Posted by dedust View Post
    differentiate :

    \ln y = 3 \ln (x+2) - 4\ln(3x+5) + \ln(\sin x) - 2\ln (2x+2)
    Taking log on both sides

    \ln y = 3 \ln (x+2) - 4\ln(3x+5) + \ln(\sin x) - 2\ln (2x+2)

    Differentiating wrt x

    \frac{1}{y} \ \frac{dy}{dx} = \frac{9}{x+2} - \frac{32}{3x+5} + cot x - \frac{8}{2x+2}

    \frac{dy}{dx} = y \ \frac{9}{x+2} - \frac{32}{3x+5} + cot x - \frac{8}{2x+2} ..................I am stuck here ??????
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  5. #5
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    Quote Originally Posted by zorro View Post
    Taking log on both sides

    \ln y = 3 \ln (x+2) - 4\ln(3x+5) + \ln(\sin x) - 2\ln (2x+2)

    Differentiating wrt x

    \frac{1}{y} \ \frac{dy}{dx} = \frac{9}{x+2} - \frac{32}{3x+5} + cot x - \frac{8}{2x+2}

    \frac{dy}{dx} = y \ \frac{9}{x+2} - \frac{32}{3x+5} + cot x - \frac{8}{2x+2} ..................I am stuck here ??????
    check again your work,..
    remember that

    \frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}

    hence
    \frac{d}{dx} \{4\ln (3x + 5) \}= \frac{4 \times 3}{(3x + 5)} = \frac{12}{3x + 5}
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  6. #6
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    I am getting the following answer

    <br />
\frac{dy}{dx} = y \frac{13}{(x-2)} - \frac{12}{(3x+5)} + cot x - \frac{4}{(2x+2)}<br />
.....................Is this correct?????
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  7. #7
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    Quote Originally Posted by zorro View Post
    I am getting the following answer

    <br />
\frac{dy}{dx} = y \frac{13}{(x-2)} - \frac{12}{(3x+5)} + cot x - \frac{4}{(2x+2)}<br />
.....................Is this correct?????
    I haven't checked all of your work but if
    [tex]\frac{1}{y}\frac{dy}{dx} = y \frac{13}{(x-2)} - \frac{12}{(3x+5)} + cot x - \frac{4}{(2x+2)}[/itex]
    then <br />
\frac{dy}{dx} = y (\frac{13}{(x-2)} - \frac{12}{(3x+5)} + cot x - \frac{4}{(2x+2)})
    (Note the parentheses.)
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  8. #8
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    thanks mite

    \frac{dy}{dx} = y \left( \frac{3}{x+2} - \frac{12}{3x+5} + cot x - \frac{4}{2x+2} \right)
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  9. #9
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    Quote Originally Posted by zorro View Post
    thanks mite

    \frac{dy}{dx} = y \left( \frac{3}{x+2} - \frac{12}{3x+5} + cot x - \frac{4}{2x+2} \right)
    don't forget to substitute back y
    \frac{dy}{dx} = \frac{(x+2)^3 (3x+5)^{-4} sinx}{(2x+2)^2} \left( \frac{3}{x+2} - \frac{12}{3x+5} + cot x - \frac{4}{2x+2} \right)
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  10. #10
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    Thanks mite
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