Question

How do I find the derivative of \[y={{e}^{ln\left( x \right)}}\]?

Answer 1

Hint: This type of problem is based on the concept of differentiation. First, we have to consider the whole function and then use chain rule \[\dfrac{d}{dx}\left( f\left( y \right) \right)={f}'\left( y \right)\dfrac{dy}{dx}\] in the given function. Therefore, we need to differentiate \[\ln \left( x \right)\] separately, that is, \[\dfrac{1}{x}\]. And then differentiate the given function with the help of chain rule, that is, multiply \[\dfrac{1}{x}\] with \[{{e}^{\ln \left( x \right)}}\] . And obtain the final answer.

Complete step by step answer:
According to the question, we are asked to find the derivative of the given function \[{{e}^{\ln \left( x \right)}}\].
We have been given the function is \[{{e}^{\ln \left( x \right)}}\] . -----(1)
First, consider \[\ln \left( x \right)\].
We have to find the differentiation of \[\ln \left( x \right)\] .
We know that \[\dfrac{d}{dx}\left( \ln \left( x \right) \right)=\dfrac{1}{x}\].
Therefore, \[\dfrac{d}{dx}\left( \ln \left( x \right) \right)=\dfrac{1}{x}\]. ---------(2)
Now, consider \[{{e}^{\ln \left( x \right)}}\].
We have to differentiate the function \[{{e}^{\ln \left( x \right)}}\] with the help of equation (2) and chain rule.
We know that \[\dfrac{d}{dx}\left( f\left( y \right) \right)={f}'\left( y \right)\dfrac{dy}{dx}\] is the chain rule.
Applying chain rule in \[{{e}^{\ln \left( x \right)}}\]
We get,
\[\Rightarrow \dfrac{d}{dx}\left( {{e}^{\ln \left( x \right)}} \right)=\dfrac{d}{dx}\left( {{e}^{\ln \left( x \right)}} \right).\dfrac{d}{dx}\left( \ln \left( x \right) \right)\] -------(3)
We know that \[\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}\].
Substitute the above result in equation (3).
We get,
\[\Rightarrow \dfrac{d}{dx}\left( {{e}^{\ln \left( x \right)}} \right)={{e}^{\ln \left( x \right)}}.\dfrac{d}{dx}\left( \ln \left( x \right) \right)\]
From equation (2), we get,
\[\dfrac{d}{dx}\left( {{e}^{\ln \left( x \right)}} \right)={{e}^{\ln \left( x \right)}}.\dfrac{1}{x}\]
Therefore, we get,
\[\dfrac{d}{dx}\left( {{e}^{\ln \left( x \right)}} \right)=\dfrac{{{e}^{\ln \left( x \right)}}}{x}\]
Hence, the derivative of \[y={{e}^{ln\left( x \right)}}\] is \[\dfrac{{{e}^{\ln \left( x \right)}}}{x}\].

Note:
Whenever you get this type of problem, we should always try to make the necessary changes in the given function to get the final solution of the function which will be the required answer. We should avoid calculation mistakes based on sign conventions. We should be thorough with the derivative of exponential and logarithmic functions and use them, if needed. Using chain rule is the only way to solve this type of question. We can also write the final solution as \[{{x}^{-1}}{{e}^{\ln \left( x \right)}}\], that is, the derivative of \[y={{e}^{ln\left( x \right)}}\] is \[{{x}^{-1}}{{e}^{\ln \left( x \right)}}\].