Evaluate $\dfrac{d}{dx}{\,\big(x^x\big)}$

Let $x$ be a variable. The variable $x$ raised to the power $x$ forms a special function $x^x$.

The derivative of the $x$-th power of $x$ with respect to $x$ is written in mathematics as follows.

$\dfrac{d}{dx}{\,\big(x^x\big)}$

The function is not an exponential function or a power function but it is a combination of both. Hence, the differentiation of this special function cannot be calculated directly and it requires some methods to find its derivative.

Proofs

The differentiation of this exponential power function can be evaluated in the following two methods.

Log method

Learn how to find the differentiation of $x$ to the $x$-th power with respect to $x$ by using logarithmic system.

Limits method

Learn how to calculate the derivative of the $x$-th power of $x$ with respect to $x$ from the first principle of differentiation.