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Math Help - basic derivatives problem

  1. #1
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    basic derivatives problem

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  2. #2
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    Hello, afeasfaerw23231233!

    \lim_{\Delta x\to0}\,\frac{(x+2)\,(x+\Delta x)^2 - x^2(x + \Delta x + 2)}{\Delta x\, \sqrt{x+2}\,\sqrt{x+\Delta x+2}\,\left[\sqrt{x+2}(x+\Delta x) + x\sqrt{x+\Delta + 2}\right]}

    . . = \;\lim_{\Delta x\to0}\,\frac{x^2+4x + x\!\cdot\!\Delta x + 2\!\cdot\!\Delta x}{\sqrt{x+2}\,\sqrt{x+\Delta x + 2}\,\left[\sqrt{x+2}(x+\Delta x) + x\sqrt{x+\Delta x + 2}\right]}
    I don't see a way to avoid expanding the numerator.
    . . It is, after all, the expected procedure.


    The numerator is: . x^3 + 2x\!\cdot\!\Delta x + x\!\cdot\!\Delta x + 2x^2 + 4x\!\cdot\!\Delta x + 2\!\cdot\!\Delta x^2 - x^3 - x^2\!\cdot\!\Delta x - 2x^2

    . . = \;x^2\!\cdot\!\Delta x + 4x\!\cdot\!\Delta x + x\!\cdot\!\Delta x + 2\!\cdot\!\Delta x^2 \;=\;\Delta x\,\left(x\!\cdot\!\Delta x + 4x + x\!\cdot\!\Delta x + 2\!\cdot\!\Delta x\right)


    Then: . \frac{\Delta x\,\left(x\!\cdot\!\Delta x + 4x + x\!\cdot\!\Delta x + 2\!\cdot\!\Delta x\right) } {{\Delta x\, \sqrt{x+2}\,\sqrt{x+\Delta x+2}\,\left[\sqrt{x+2}(x+\Delta x) + x\sqrt{x+\Delta + 2}\right]}}

    . . . . = \;\frac{x^2+4x + x\!\cdot\!\Delta x + 2\!\cdot\!\Delta x}{\sqrt{x+2}\,\sqrt{x+\Delta x + 2}\,\left[\sqrt{x+2}\left(x + \Delta x\right) + x\sqrt{x + \Delta x + 2}\right]}

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  3. #3
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    thanks, but it takes me so much time to expand this piece

    !!!
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