How do you differentiate x ^ x ?

Book A Demo Class

Derivative in mathematics is the rate of change of a function with respect to a variable.

Derivatives are very important to the solution of problems in calculus. To differentiate a function one must know the rules of derivatives.

The derivative of a constant function is always zero.

Let $y\,=\,f\,(x)$ is any function.

We can explain it as the measure of the rate at which the value of changes with respect to the change of the variable $x$ . It is read as the derivative of function $f$ with respect to the variable $x$ . Now the question is how we can differentiate $\,{{x}^{x}}?$

So, regarding this question we can say that there are two ways to find the derivative of ${{x}^{x}}$ .

There is an important point to be noticed here that this function is neither a power function of the form $x{{\,}^{\hat{\ }}}\,k$ where $k$ is any constant nor an exponential function of the form $b\,\hat{\ }x$ where b is a constant, so we can’t use the differentiation formulas for either of these cases directly.

Symbol of derivative

The symbol for denoting the derivative of a function is $f'(x)$ or we can also denote it by $\frac{{d(y)}}{{dx}}$

Differentiating $y\,=\,x\,\hat{\ }\,x$ is simple but tricky.

Method 1

The best way to solve it is to bring the power down and we can bring the power down only by taking logarithm (ln) to the natural base e on both sides.

$Let\,y\,=\,x\,\hat{\ }\,x$

1Taking log both sides ,we get0

ln\,y\,=\,ln\,x

Now taking derivative on both sides, we get

\frac{{d(\ln \,y)}}{{dx}}\,=\,\frac{{d\,(x\,\ln \,x)}}{{dx}}

To differentiate the right hand side, we can use the product rule

Product rule says

First function $d/dx$ second function + second function $d/dx$ first function
Also, According to properties of derivatives: derivative of $\log \,x$ is $1/x$ and derivative of $x$ is 1.

So, applying this rule above we get

\begin{array}{l}1/y\,\,\frac{{dy}}{{dx}}\,\,=\,x\,\frac{d}{{dx}}\,(ln\,x)\,+\,ln\,x\,\frac{d}{{dx}}\,(x)\\1/y\,\,\frac{{dy}}{{dx}}\,\,=\,\,x(1/x)\,\,+\,\,ln\,x\,(1)\\1/y\,\,\frac{{dy}}{2}\,\,=\,\,1+\,ln\,x\\\frac{{dy}}{{dx}}\,\,=\,y\,(1\,+\,ln\,x)\\\frac{{dy}}{{dx}}\,\,=\,\,x\hat{\ }x\,(1+ln\,x)\,\,\,[as\,the\,value\,of\,y\,=\,x\,\hat{\ }\,x]\end{array}

So , derivative of ${{x}^{x}}\,is\,{{x}^{x}}(1\,+\,ln\,x)$ by applying logarithm method as this can’t be differentiated with the usual differential methods of derivatives such as product rule, chain rule, quotient rule etc.