In this post, I will show you the steps to find the derivative of function \(\displaystyle f(x) = x^x \)
Steps to find the derivative:
\(
f(x) = x^x \\
\Rightarrow \ln(f(x)) = \ln(x^x) \scriptsize \text{, take the natural logarithm of both sides} \\
\Rightarrow \ln(f(x)) = x \cdot \ln(x) \\
\displaystyle
\Rightarrow \frac{d}{dx}[\ln(f(x))] = \frac{d}{dx}[x \cdot \ln(x)] \scriptsize \text{, differentiate both sides with respect to } x \\
\displaystyle
\Rightarrow \frac{1}{f(x)} \cdot \frac{d}{dx}f(x) = \ln(x) + x \cdot \frac{1}{x} \scriptsize \text{, use the chain rule on the left side and the product rule on the right side} \\
\displaystyle
\Rightarrow \frac{1}{f(x)} \cdot \frac{d}{dx}f(x) = \ln(x) + 1 \\
\displaystyle
\Rightarrow \frac{d}{dx}f(x) = f(x) \cdot (\ln(x) + 1) \scriptsize \text{, multiply both sides by } f(x) \\
\displaystyle
\Rightarrow \frac{d}{dx}f(x) = x^x (\ln(x) + 1) \\ \\
\displaystyle
\text{Therefore, } \fbox{$\frac{d}{dx}(x^x) = x^x (\ln(x) + 1)$}
\)
