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Math Help - derivativ

  1. #1
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    derivativ

    find f prim (1) when f(x)= x^1+sqrt x .

    thanks
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  2. #2
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    Hi

    f(x) = x^{1+\sqrt{x}} = e^{(1+\sqrt{x})ln(x)}

    The first derivative of

    e^{u(x)} is u'(x) e^{u(x)}

    Here u(x) = (1+\sqrt{x})ln(x)

    u(x) is a product of 2 functions
    u'(x) = \frac{1}{2 \sqrt{x}}ln(x) + \frac{1+\sqrt{x}}{x}
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  3. #3
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    hi !
    I dont know this is power fuction or exponetial function can you explain more .
    thank you
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  4. #4
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    familiar with logarithmic differentiation?

    y = x^{1+\sqrt{x}}

    \ln{y} = \ln\left(x^{1+\sqrt{x}}\right)

    \ln{y} = (1 + \sqrt{x})\ln{x}<br />

    \frac{d}{dx}[\ln{y} = (1 + \sqrt{x})\ln{x}]

    \frac{y'}{y} = (1 + \sqrt{x})\cdot \frac{1}{x} + \ln{x} \cdot \frac{1}{2\sqrt{x}}

    y' = y\left((1 + \sqrt{x})\cdot \frac{1}{x} + \ln{x} \cdot \frac{1}{2\sqrt{x}}\right)

    y' = x^{1+\sqrt{x}}\left((1 + \sqrt{x})\cdot \frac{1}{x} + \ln{x} \cdot \frac{1}{2\sqrt{x}}\right)
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