Derivative with Logarithmic Differentiation

Problem 109: Find the derivative of $f(x)$ using logarithmic differentiation when $f(x) = x^{e^x}$ .

Solution: We will take the derivative of $f(x)$ using logarithmic properties. That is,

(1) $\begin{align*} f(x) & = x^{e^x} \\ \ln(f(x)) & = \ln(x^{e^x}) \\ \ln(f(x)) & = e^x \cdot \ln(x) \quad \text{(property of logarithm)}. \end{align*}$

Differentiating both side we get

(2) $\begin{align*}\frac{1}{f(x)} \cdot f'(x) & = e^x \cdot \ln(x) + \frac{e^x}{x} \\ f'(x) & = f(x) \cdot \left(e^x \cdot \ln(x) + \frac{e^x}{x} \right) \\ & = x^{e^x} \cdot \left(e^x \cdot \ln(x) + \frac{e^x}{x} \right) . \end{align*}$

That is,

(3) $\begin{equation*} \boxed{f'(x) = x^{e^x} \cdot \left(e^x \cdot \ln(x) + \frac{e^x}{x} \right)}. \end{equation*}$

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Derivative with Logarithmic Differentiation

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