# Calculating Derivative – A function to the power of a function – Exercise 6374

Exercise

Find the derivative of the following function:

$$f(x)=x^{x^2}$$

$$f'(x)=x^{x^2+1}\cdot (2\ln x+1)$$

Solution

We do not have a derivative formula for a function to the power of a function. To work around this, we use a “trick” – we use logarithm rules to get a multiplication of functions instead of a function to the power of a function.

$$f(x)=x^{x^2}=$$

$$=e^{\ln x^{x^2}}=$$

$$=e^{x^2\ln x}$$

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

$$f'(x)=e^{x^2\ln x}\cdot (2x\ln x+x^2\cdot\frac{1}{x})=$$

One can simplify the derivative:

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

By logarithm rules we get:

$$=x^{x^2}\cdot (2x\ln x+x)=$$

$$=x^{x^2+1}\cdot (2\ln x+1)$$

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