What is the derivative of x^{e^{x^2}} ?

The derivative of x^{e^{x^2}} is x^{e^{x^2}}(2e^{x^2}xln(x)+(e^{x^2})/x)

What is the first derivative of x^{e^{x^2}} ?

The first derivative of x^{e^{x^2}} is x^{e^{x^2}}(2e^{x^2}xln(x)+(e^{x^2})/x)

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\frac{d}{d x} (x^{e^{x^{2}}})

x^{e^{x^{2}}} (2 e^{x^{2}} x ln (x) + \frac{e ^{x^{2}}}{x})

Solution steps

\frac{d}{d x} (x^{e^{x^{2}}})

Apply exponent rule:

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

x^{e^{x^{2}}} = e^{e^{x^{2}} l n (x)}

= \frac{d}{d x} (e^{e^{x^{2}} l n (x)})

Apply the chain rule:

e^{e^{x^{2}} l n (x)} \frac{d}{d x} (e^{x^{2}} ln (x))

= e^{e^{x^{2}} l n (x)} \frac{d}{d x} (e^{x^{2}} ln (x))

\frac{d}{d x} (e^{x^{2}} ln (x)) = 2 e^{x^{2}} x ln (x) + \frac{e ^{x^{2}}}{x}

= e^{e^{x^{2}} l n (x)} (2 e^{x^{2}} x ln (x) + \frac{e ^{x^{2}}}{x})

e^{e^{x^{2}} l n (x)} = x^{e^{x^{2}}}

= x^{e^{x^{2}}} (2 e^{x^{2}} x ln (x) + \frac{e ^{x^{2}}}{x})

What is the derivative of x^{e^{x^2}} ?
The derivative of x^{e^{x^2}} is x^{e^{x^2}}(2e^{x^2}xln(x)+(e^{x^2})/x)
What is the first derivative of x^{e^{x^2}} ?
The first derivative of x^{e^{x^2}} is x^{e^{x^2}}(2e^{x^2}xln(x)+(e^{x^2})/x)