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

Show final answer

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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