Logarithmic Differentiation:
\(\displaystyle y = x^{\sqrt{x}}\)
\(\displaystyle y = x^{x^{1/2}}\)
\(\displaystyle \ln(y) = \ln(x^{x^{1/2}})\)
\(\displaystyle \ln(y) = x^{1/2}\ln x\)
\(\displaystyle \dfrac{1}{y}y' = \dfrac{1}{2}x^{-1/2}\dfrac{1}{x}\)
\(\displaystyle y' = (y) \dfrac{1}{2}x^{-1/2}\dfrac{1}{x}\)
\(\displaystyle y' = (x^{\sqrt{x}}) \dfrac{1}{2}x^{-1/2}\dfrac{1}{x}\)
Is this right, or on the right track?
\(\displaystyle y = x^{\sqrt{x}}\)
\(\displaystyle y = x^{x^{1/2}}\)
\(\displaystyle \ln(y) = \ln(x^{x^{1/2}})\)
\(\displaystyle \ln(y) = x^{1/2}\ln x\)
\(\displaystyle \dfrac{1}{y}y' = \dfrac{1}{2}x^{-1/2}\dfrac{1}{x}\)
\(\displaystyle y' = (y) \dfrac{1}{2}x^{-1/2}\dfrac{1}{x}\)
\(\displaystyle y' = (x^{\sqrt{x}}) \dfrac{1}{2}x^{-1/2}\dfrac{1}{x}\)
Is this right, or on the right track?
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