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

Exercise

Find the derivative of the following function:

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

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

Solution

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

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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