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

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

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

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

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derivative of

f (x) = x^{e^{x}}

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

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{e^{x}} = e^{e^{x} l n (x)}

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

Apply the chain rule:

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

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

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

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

e^{e^{x} l n (x)} = x^{e^{x}}

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

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