Note the following properties of Natural Logarithms.
Solution:
Apply the natural logarithm to both sides of this equation:
\[y = x^{e^{x}} \]
\[ln(y) = ln(x^{e^{x}}) \]
\[ln(y) = e^{x} ln(x) \]
Differentiate both sides of this equation. The lefthand side requires the chain rule since y represents a function of x . Use the product rule on the righthand side. We get:
\[\frac{1}{y} dy/dx = e^{x} d/dx (ln(x) ) + d/dx(e^{x} ln(x)\]
\[\frac{1}{y} dy/dx = \frac{1}{x} e^{x} + e^{x} ln(x) \]
Get a common denominator and combine fractions on the righthand side.
\[ = \frac{e^{x}}{x} + \frac{x}{x}e^{x}lnx \]
\[ = \frac{e^{x} + x e^{x}lnx}{x} \]
Factor out e^x.
\[ = \frac{e^{x}(1 + xlnx)}{x}\]
Multiply both sides of this equation by y:
\[ \frac{dy}{dx} = y\frac{e^{x}(1 + xlnx)}{x}\]
\[ \frac{dy}{dx} = x^{e^x}\frac{e^{x}(1 + xlnx)}{x}\]
\[ \frac{dy}{dx} = x^{e^x}\frac{e^{x}(1 + xlnx)}{x^1}\]
\[ \frac{dy}{dx} = x^{e^{x  1}}e^{x}(1 + xlnx)\]
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