Calculating Derivative – e to the power of a multiplication of x and ln – Exercise 6369


Find the derivative of the following function:

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

Final Answer

f'(x)=x^x\cdot (\ln x+1)


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

Using Derivative formulas and the multiplication rule and chain rule in Derivative Rules, we get the derivative:

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

One can simplify the derivative using logarithm rules:

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

=x^x\cdot (\ln x+1)

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