Question

For \[f\left( x \right) = {\log _{{e^n}}}{x^n}\] then find \[f'\left( 1 \right)\].

Answer 1

Hint: Here we have to find the value of the derivative of the given function. We will first find the derivative of the given function with respect to the required variable. Then we will substitute the value of the variable in the obtained derivative of the function to get the required answer.

Complete step-by-step answer:
Here we have to find the value of the derivative of the given function.
The given function is \[f\left( x \right) = {\log _{{e^n}}}{x^n}\].
We know from the properties of the logarithm function that \[{\log _a}{b^n} = n{\log _a}b\]
Now, we will use the same property of the logarithmic function in the given function. Therefore, we get
\[ \Rightarrow f\left( x \right) = n{\log _{{e^n}}}x\]
Using the property of the logarithmic function \[{\log _{{a^n}}}b = \dfrac{1}{n}{\log _a}b\], we get
\[ \Rightarrow f\left( x \right) = \dfrac{n}{n}{\log _e}x\]
On further simplifying the terms, we get
\[ \Rightarrow f\left( x \right) = {\log _e}x\]
Now, we will differentiate both sides with respect to the variable \[x\]. Therefore, we get
\[ \Rightarrow \dfrac{{d\left( {f\left( x \right)} \right)}}{{dx}} = \dfrac{{d\left( {{{\log }_e}x} \right)}}{{dx}}\]
\[ \Rightarrow f'\left( x \right) = \dfrac{1}{x}\]
Now, we will replace the value of variable \[x\] with 1, Therefore
 \[ \Rightarrow f'\left( 1 \right) = \dfrac{1}{1} = 1\]
Hence, the required value of the derivative of the given function is equal to 1.

Note: Here we have used various properties of the logarithmic function. A logarithmic function is defined as the function which is the inverse of the exponential function. We need to remember that the value of the logarithm of the negative numbers is not defined. The logarithm of any positive number, whose base is a number, which is greater than zero and is not equal to the number one, is the power to which the base can be raised in order to obtain the given number.