Answer
397.5k+ views
Hint: Differentiation is the rate change of a function with respect to the variable on which the function depends. Here, the given question can differentiate the given function in two different ways. One by chain rule and the other way by quotient rule.
Complete step by step solution:
Let us first understand what is meant by differentiation.Differentiation in mathematics can be defined in many ways. Differentiation is the rate change of a function with respect to the variable on which the function depends. If we draw a graph of the function with respect to the independent variable, then the differentiation of the function at any point is the equal to the slope of the tangent to the curve of the function at that point. If the function is f(x) then differentiation of f(x) is written as $\dfrac{d}{dx}f(x)$.
Here, the given function is $f(x)=\dfrac{1}{\ln x}$
Therefore, differentiation of $\dfrac{1}{\ln x}$ is $\dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}$.
By using chain rule, we get that
$\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}=-1{{\left( \ln x \right)}^{-2}}.\dfrac{d}{dx}\ln x$ …… (i)
The differentiation of ln(x) is equal to $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$.
Substitute this value in equation (i).
Then we get that,
$\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}=-1{{\left( \ln x \right)}^{-2}}.\dfrac{1}{x}$
Therefore, on further simplification we get that
$\therefore\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}=\dfrac{-1}{x{{\left( \ln x \right)}^{2}}}$.
Note:For the differentiation of the given function $f(x)=\dfrac{1}{\ln x}$ we can also use the product rule or quotient rule of differentiation. The product rule for the differentiation of a function $f(x)=g(x)h(x)$ is given as,
$\dfrac{d}{dx}f(x)=\left[ \dfrac{d}{dx}g(x) \right]h(x)+g(x)\left[ \dfrac{d}{dx}h(x) \right]$
The quotient rule for the differentiation of a function $f(x)=\dfrac{g(x)}{h(x)}$ is given as
$\dfrac{d}{dx}f(x)=\dfrac{\left[ \dfrac{d}{dx}g(x) \right]h(x)-g(x)\left[ \dfrac{d}{dx}h(x) \right]}{{{\left[ h(x) \right]}^{2}}}$
Therefore, we can use the quotient rule for the given function to differentiate as:
$\Rightarrow \dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{\left[ \dfrac{d}{dx}(1) \right]\ln x-1.\left[ \dfrac{d}{dx}\ln x \right]}{{{\left( \ln x \right)}^{2}}}$
$\Rightarrow \dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{0-\left[ \dfrac{1}{x} \right]}{{{\left( \ln x \right)}^{2}}}$
$\therefore \dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{-1}{x{{\left( \ln x \right)}^{2}}}$
Complete step by step solution:
Let us first understand what is meant by differentiation.Differentiation in mathematics can be defined in many ways. Differentiation is the rate change of a function with respect to the variable on which the function depends. If we draw a graph of the function with respect to the independent variable, then the differentiation of the function at any point is the equal to the slope of the tangent to the curve of the function at that point. If the function is f(x) then differentiation of f(x) is written as $\dfrac{d}{dx}f(x)$.
Here, the given function is $f(x)=\dfrac{1}{\ln x}$
Therefore, differentiation of $\dfrac{1}{\ln x}$ is $\dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}$.
By using chain rule, we get that
$\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}=-1{{\left( \ln x \right)}^{-2}}.\dfrac{d}{dx}\ln x$ …… (i)
The differentiation of ln(x) is equal to $\dfrac{d}{dx}\ln x=\dfrac{1}{x}$.
Substitute this value in equation (i).
Then we get that,
$\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}=-1{{\left( \ln x \right)}^{-2}}.\dfrac{1}{x}$
Therefore, on further simplification we get that
$\therefore\dfrac{d}{dx}{{\left( \ln x \right)}^{-1}}=\dfrac{-1}{x{{\left( \ln x \right)}^{2}}}$.
Note:For the differentiation of the given function $f(x)=\dfrac{1}{\ln x}$ we can also use the product rule or quotient rule of differentiation. The product rule for the differentiation of a function $f(x)=g(x)h(x)$ is given as,
$\dfrac{d}{dx}f(x)=\left[ \dfrac{d}{dx}g(x) \right]h(x)+g(x)\left[ \dfrac{d}{dx}h(x) \right]$
The quotient rule for the differentiation of a function $f(x)=\dfrac{g(x)}{h(x)}$ is given as
$\dfrac{d}{dx}f(x)=\dfrac{\left[ \dfrac{d}{dx}g(x) \right]h(x)-g(x)\left[ \dfrac{d}{dx}h(x) \right]}{{{\left[ h(x) \right]}^{2}}}$
Therefore, we can use the quotient rule for the given function to differentiate as:
$\Rightarrow \dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{\left[ \dfrac{d}{dx}(1) \right]\ln x-1.\left[ \dfrac{d}{dx}\ln x \right]}{{{\left( \ln x \right)}^{2}}}$
$\Rightarrow \dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{0-\left[ \dfrac{1}{x} \right]}{{{\left( \ln x \right)}^{2}}}$
$\therefore \dfrac{d}{dx}\left( \dfrac{1}{\ln x} \right)=\dfrac{-1}{x{{\left( \ln x \right)}^{2}}}$
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)