Math - Derivatives

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Differentiate the given function. Simplify the answer.

f(x) = 28x*ln√x + 42

f'(x) = 28*ln(x)^(1/2) + (1/(x^(1/2))*(1/2(x)^(-1/2)28x)

f'(x) = 28*ln(x)^(1/2) + (14x*x^(-1/2))/x^(1/2)

f'(x) = 28*ln(x)^(1/2) + (14x^(-1/2))/x^(1/2)

f'(x) = 28*ln(x)^(1/2) + (14x^(-1)

f'(x) = 28*ln(x)^(1/2) + 1/14x

... how do I simplify further to "14(1 + lnx)"??

  • Math - Derivatives -

    28 x ln [sqrt (x+42)] ?maybe? I need more parentheses to understand.

    [28 x/sqrt(x+42) ](1/2)x/sqrt(x+42) +28 ln(sqrt(x+42)]

    14 x^2/(x+42) + 28 ln[sqrt(x+42]

  • Math - Derivatives -

    The original equation is...

    f(x) = (28x)(ln(sqrt(x)) + 42

  • Math - Derivatives -

    Well then, the 42 is irrelevant for our purposes since its derivative is zero.
    so
    28 x (d/dx ln x^.5) + 28 ln x^.5
    28 x [ (1/x^.5) .5 x^-.5 ] + 28 ln x^.5
    28 x (.5/x) + 28 ln x^.5
    14 + 28 ln sqrt x

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