Question

How do you find the derivative of \[\ln {{\left( x \right)}^{\dfrac{1}{2}}}\]?

Answer 1

Hint: In the given question, we have been asked to differentiate the logarithmic function. The given equation is based on the concepts of derivation of the natural log. In order to differentiate the given function, we will apply the identities of logarithmic function to simplify the given expression i.e. \[\log {{a}^{b}}=b\log a\]. Later as we know that the natural derivative of logarithmic function or log is\[\dfrac{d}{dx}\ln x=\dfrac{1}{x}\], thus differentiating the given expression. In this way we will get the required solution or the derivative of the given function or expression.

Formula used:
The natural derivative of logarithmic function or log is given by;
\[\dfrac{d}{dx}\ln x=\dfrac{1}{x}\]
Using the properties of logarithmic function;
\[\log {{a}^{b}}=b\log a\]

Complete step by step solution:
We have given that,
\[\Rightarrow \ln {{\left( x \right)}^{\dfrac{1}{2}}}\]
Using the properties of logarithmic function;
\[\log {{a}^{b}}=b\log a\]
As we know that,
The natural derivative of logarithmic function or log is given by;
\[\dfrac{d}{dx}\ln x=\dfrac{1}{x}\]
Therefore,
We have,
\[\Rightarrow \dfrac{d}{dx}\ln {{\left( x \right)}^{\dfrac{1}{2}}}=\dfrac{d}{dx}\left( \dfrac{1}{2}\ln x \right)\]
Taking out the constant part from the derivative,
\[\Rightarrow \dfrac{d}{dx}\ln {{\left( x \right)}^{\dfrac{1}{2}}}=\dfrac{1}{2}\dfrac{d}{dx}\left( \ln x \right)\]
Using the derivative of natural log i.e. \[\dfrac{d}{dx}\ln x=\dfrac{1}{x}\]
\[\Rightarrow \dfrac{d}{dx}\ln {{\left( x \right)}^{\dfrac{1}{2}}}=\dfrac{1}{2}\cdot \dfrac{1}{x}\]
Simplifying the above, we will get
\[\Rightarrow \dfrac{d}{dx}\ln {{\left( x \right)}^{\dfrac{1}{2}}}=\dfrac{1}{2x}\]

Therefore, the derivative of \[\ln {{\left( x \right)}^{\dfrac{1}{2}}}\] is \[\dfrac{1}{2x}\].

Note: While solving this question, students should always know the basic standard deviation formulas as we can use these formulas directly.

For example: The natural derivative of logarithmic function or log is \[\dfrac{d}{dx}\ln x=\dfrac{1}{x}\].
Students should know the basic properties of logarithmic function to simplify the given expression in the question. We should perform each step carefully in order to avoid confusion otherwise it will result in making errors.