Logarithmic Differentiation - Problem # 2

[Jason76]

Logarithmic Differentiation:

\(\displaystyle y = x^{e^{x}}\)

\(\displaystyle \ln(y) = \ln(x^{e^{x}})\)

\(\displaystyle \ln(y)= e^{x}\ln x\)

\(\displaystyle \dfrac{1}{y}y' = e^{x} \dfrac{1}{x}\)

\(\displaystyle y' = (y) e^{x} \dfrac{1}{x}\)

\(\displaystyle y' = (x^{e^{x}}) e^{x} \dfrac{1}{x}\) Right or on the right track?

 

[stapel]

Logarithmic Differentiation:

\(\displaystyle y = x^{e^{x}}\)

\(\displaystyle \ln(y) = \ln(x^{e^{x}})\)

\(\displaystyle \ln(y)= e^{x}\ln x\)

\(\displaystyle \dfrac{1}{y}y' = e^{x} \dfrac{1}{x}\)

Shouldn't the Product Rule spit out two terms for the derivative?

 

[HallsofIvy]

As stapel suggested, you have differentiated the product \(\displaystyle e^x ln(x)\) incorrectly. The product rule gives
(fg)'= f'g+ fg', NOT f' g'.