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Given function : y = (lnx)^x²

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Apply log each side with base e.

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lny = x²ln (lnx)

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Apply derivative with respect to x

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(d/dx)(lny) = (d/dx)(x²ln(lnx))

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Apply formulas : (d/dx)(UV)= V(dU/dx)+U(dV/dx) and (d/dx)(lnU) = (1/U)(dU/dx)

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Here consider U = x² and V = ln (lnx)

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(1/y)(dy/dx) =(ln(lnx))(d/dx)(x²)+(x²)(d/dx)(ln(lnx))

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Apply formulas : (d/dx)(f)ⁿ = (nf^n-1)(df/dx)  and   (d/dx)(lnU) = (1/U)(dU/dx)

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(1/y)(dy/dx) =(ln(lnx))(2x)+(x²)(1/lnx) (d/dx)(lnx)

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Apply formula : (d/dx)(lnx) = 1/x

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(1/y)(dy/dx) =2xln(lnx)+(x²)(1/lnx) (1/x)

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(dy/dx) =y [ 2xln(lnx)+(x)(1/lnx)]

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Substitute y = (lnx)^x²

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(dy/dx) =(lnx)^x²[ 2xln(lnx)+(x/lnx) ]

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The solution is  (dy/dx) =(lnx)^x²[ 2xln(lnx)+(x/lnx) ].

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