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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}

Final Answer


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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