College Calc

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Problem: y=sqrtx ^1/x. Find DY/DX.

ln(f(x))=lnx^1/2x or lnx to the 1/2x

1/2x lnx

product rule

(1/4x^2)(lnx)+(1/x)(1/2x)
is the product rule correct?
What do i do after this product rule?

  • College Calc -

    y = √(x^(1/x)) = x^(1/2x)
    ln y = 1/(2x) ln x
    1/y y' = -1/(2x^2) lnx + 1/(2x)(1/x)
    1/y y' = (1-lnx)/(2x^2)
    y' = (1-lnx)y/(2x^2)
    = (1-lnx) √(x^(1/x))/(2x^2)

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