Question

Does the exponential rule apply to the function $f\left(x\right)=x^x$? What about the power rule? Explain your answers.

Short answer

We cannot use the exponential rule because to apply it base should be constant and exponent should be variable

We cannot use the power rule because to apply it when base is variable and exponent should be constant

After differentiation we get :$\frac{dy}{dx}=x^x(1+\ln x)$

Step-by-step solution

. Given Information

We are given :$f\left(x\right)=x^x$

Step 2. Exponential rule

We will not be able to use the exponential rule because to use it base should be constant and exponent should be variable but here the base is not constant

Step 3. Power rule

We will not be able to use the power rule because to use it the base should be variable and exponent should be constant but here the exponent is not constant

Step 4. Differentiating

Let $y=x^x$

Taking logarithm both sides :

$$\begin{gathered} \ln \left(y\right)=\ln x^x \\ \ln \left(y\right)=x\ln x \end{gathered}$$

Now we differentiate with respect to x :

$$\begin{gathered} \frac{1}{y}.\frac{dy}{dx}=x\frac{1}{x}+\ln x \\ \frac{1}{y}.\frac{dy}{dx}=1+\ln x \\ \frac{dy}{dx}=y(1+\ln x) \\ \frac{dy}{dx}=x^x(1+\ln x) \end{gathered}$$