How do you differentiate $f\left( x \right)={{e}^{x}}\ln x$?

Hint: We first define the multiplication rule and how the differentiation of function works. We take addition of these two different differentiated values. We take the $\dfrac{dy}{dx}$ altogether. We keep one function and differentiate the other one and then do the same thing with the other function. Then we take the addition to complete the formula.

Complete step-by-step solution:
We now discuss the multiplication process of two functions where \[f\left( x \right)=u\left( x \right)v\left( x \right)\]
Differentiating \[f\left( x \right)=uv\], we get \[\dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{d}{dx}\left[ uv \right]=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\].
The above-mentioned rule is the multiplication rule. We apply that on $f\left( x \right)={{e}^{x}}\ln x$. We assume the functions where \[u\left( x \right)={{e}^{x}},v\left( x \right)=\ln x\]
We know that differentiation of \[u\left( x \right)={{e}^{x}}\] is ${{u}^{'}}\left( x \right)={{e}^{x}}$ and differentiation of \[v\left( x \right)=\ln x\] is \[{{v}^{'}}\left( x \right)=\dfrac{1}{x}\]. We now take differentiation on both parts of $f\left( x \right)={{e}^{x}}\ln x$ and get \[\dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{d}{dx}\left[ {{e}^{x}}\ln x \right]\].
We place the values of ${{u}^{'}}\left( x \right)={{e}^{x}}$ and \[{{v}^{'}}\left( x \right)=\dfrac{1}{x}\] to get
\[\dfrac{d}{dx}\left[ {{e}^{x}}\ln x \right]={{e}^{x}}\dfrac{d}{dx}\left( \ln x \right)+\left( \ln x \right)\dfrac{d}{dx}\left( {{e}^{x}} \right)\].
We take all the $\dfrac{dy}{dx}$ forms altogether to get
\[\begin{align}
  & \dfrac{d}{dx}\left[ f\left( x \right) \right]=\dfrac{d}{dx}\left[ {{e}^{x}}\ln x \right] \\
 & \Rightarrow {{f}^{'}}\left( x \right)={{e}^{x}}\times \dfrac{1}{x}+\left( \ln x \right)\left( {{e}^{x}} \right)=\dfrac{{{e}^{x}}}{x}+{{e}^{x}}\ln x \\
 & \Rightarrow {{f}^{'}}\left( x \right)={{e}^{x}}\left( \ln x+\dfrac{1}{x} \right) \\
\end{align}\]
Therefore, differentiation of $f\left( x \right)={{e}^{x}}\ln x$ is \[{{e}^{x}}\left( \ln x+\dfrac{1}{x} \right)\].

Note: We need to remember that in the chain rule \[\dfrac{d}{d\left[ h\left( x \right) \right]}\left[ goh\left( x \right) \right]\times \dfrac{d\left[ h\left( x \right) \right]}{dx}\], we aren’t cancelling out the part \[d\left[ h\left( x \right) \right]\]. Cancelation of the base differentiation is never possible. It’s just a notation to understand the function which is used as a base to differentiate. The rule may be extended or generalized to many other situations, including to products of multiple functions, to a rule for higher-order derivatives of a product, and to other contexts.