What is the derivative of f(x)=e^xln(x) ?

The derivative of f(x)=e^xln(x) is e^xln(x)+(e^x)/x

What is the first derivative of f(x)=e^xln(x) ?

The first derivative of f(x)=e^xln(x) is e^xln(x)+(e^x)/x

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derivative

e^{x} ln (x) + \frac{e ^{x}}{x}

Solution steps

\frac{d}{d x} (e^{x} ln (x))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{x}, g = ln (x)

= \frac{d}{d x} (e^{x}) ln (x) + \frac{d}{d x} (ln (x)) e^{x}

\frac{d}{d x} (e^{x}) = e^{x}

\frac{d}{d x} (ln (x)) = \frac{1}{x}

= e^{x} ln (x) + \frac{1}{x} e^{x}

\frac{1}{x} e^{x} = \frac{e ^{x}}{x}

= e^{x} ln (x) + \frac{e ^{x}}{x}

What is the derivative of f(x)=e^xln(x) ?
The derivative of f(x)=e^xln(x) is e^xln(x)+(e^x)/x
What is the first derivative of f(x)=e^xln(x) ?
The first derivative of f(x)=e^xln(x) is e^xln(x)+(e^x)/x