Question

What is the domain of the derivative of $\ln x$ ?

Answer 1

Domain of a function is the set of all the input values upon which the function is defined. In this, we need to find the domain of the derivative of $\ln x$. For that, we will first find the derivative of $\ln x$. After that, we will try to find the values for which the derivative of $\ln x$ is not defined. We will then subtract the points for which the derivative is not defined from the set of real numbers to find the domain of that function.

Complete step by step answer:
We need to find the domain of the derivative of $\ln x$.
First of all, we will find the derivative of $\ln x$
Let $y = \ln x$. So, we have to find $\dfrac{{dy}}{{dx}}$.
Differentiating both sides of $y = \ln x$, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\ln x} \right)$
Using $\dfrac{d}{{dx}}\left( {\ln x} \right) = \dfrac{1}{x}$, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\ln x} \right) = \dfrac{1}{x}$
Hence, we get
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{x}$
Hence, the derivative of $\ln x$ is $\dfrac{1}{x}$.

Now, we need to find the domain of derivatives of $\ln x$. i.e. we need to find the domain of $\dfrac{1}{x}$.
Let us first find the points where $\dfrac{1}{x}$ is not defined.
We know that a fraction $\dfrac{a}{b}$ is not defined on the points where $b = 0$.
Hence, $\dfrac{1}{x}$ is not defined for $x = 0$.
So, we see that $\dfrac{1}{x}$ is not defined only when $x = 0$ and is defined for all the other real values of $x$.
Hence, Domain of $\dfrac{1}{x}$ is \mathbb{R} - \left\\{ 0 \right\\}, where $\mathbb{R}$ is the set of all real numbers.

Therefore, we get, Domain of the derivative of $\ln x$ is \mathbb{R} - \left\\{ 0 \right\\}, where $\mathbb{R}$ is the set of all real numbers.

Note: To find the domain of a fraction $\dfrac{{f\left( x \right)}}{{g\left( x \right)}}$, we usually solve $g\left( x \right) = 0$ and then delete all the values of $x$ which satisfies $g\left( x \right) = 0$ from the set of Real Numbers. In case we forget the formula for the derivative of $\ln x$, we can use the First Principle of Differentiation. According to the First Principle of Differentiation, $f'\left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$, where $f'\left( x \right)$ is the derivative of the function $f\left( x \right)$ with respect to $x$.