Question

How do you find the derivative of $y = \ln {e^x}$.

Answer 1

Hint: In order to determine the derivative of the above question with respect to variable $x$, Apply the chain rule as we don’t have a single variable but we have a function as $X = {e^x}$. So According to Chain rule $\dfrac{d}{{dx}}(\ln X) = \dfrac{1}{X}.\dfrac{d}{{dx}}(X)$. Now using this rule and using the another rule of derivative $\dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$, you will obtain your desired result.

Formula used:
$ \Rightarrow \dfrac{d}{{dx}}(\ln x) = \dfrac{1}{x}$
$ \Rightarrow \dfrac{d}{{dx}}({e^x}) = {e^x}$

Complete step by step answer:
We are Given a expression $y = \ln {e^x}$ and we have to find the derivative of this expression with respect to x.
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(\ln {e^x})$
Let’s Assume ${e^x}$ as X
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(\ln X)$
Now Applying Chain rule to the above derivative which says that if we are not given a single variable $x$ and instead of it a function is given($X$) with the logarithm then the derivative will become
$ \Rightarrow \dfrac{d}{{dx}}(\ln X) = \dfrac{1}{X}.\dfrac{d}{{dx}}(X)$
We know that Derivative of $\ln x$ is equal to $\dfrac{1}{x}$
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{X}.\dfrac{d}{{dx}}(X)$
Putting back $X$
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{{e^x}}}.\dfrac{d}{{dx}}({e^x})$
Since $\dfrac{d}{{dx}}({e^x}) = {e^x}$
$
\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{{{e^x}}}.({e^x}) \\
\Rightarrow \dfrac{{dy}}{{dx}} = 1 \\
 $
Therefore , the derivative of $y = \ln {e^x}$ with respect to x is equal to 1.

Additional Information:
1. What is Differentiation?
It is a method by which we can find the derivative of the function .It is a process through which we can find the instantaneous rate of change in a function based on one of its variables.
Let y = f(x) be a function of x. So the rate of change of $y$ per unit change in $x$ is given by:
$\dfrac{{dy}}{{dx}}$.
2. Value of constant ‘e’ is equal to $2.71828$.
3. A logarithm is basically the reverse of a power or we can say when we calculate a logarithm of any number, we actually undo an exponentiation.
4. Any multiplication inside the logarithm can be transformed into addition of two separate logarithm values.
${\log _b}(mn) = {\log _b}(m) + {\log _b}(n)$
5. Any division inside the logarithm can be transformed into subtraction of two separate logarithm values.
${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}(m) - {\log _b}(n)$
6. Any exponent value on anything inside the logarithm can be transformed and moved out of the logarithm as a multiplier and vice versa.
$n\log m = \log {m^n}$

Note: Logarithm is basically the inverse of exponent. Similarly differentiation is the inverse of integration. Chain rule is applied only when there is a need to differentiate a function not a single variable.