Answer
424.5k+ views
Hint: Start by finding the derivative of $ \sec \left( {{\tan }^{-1}}x \right) $ using the chain rule. We know that $ \dfrac{d\left( \sec x \right)}{dx}=\sec x\tan x $ and $ \dfrac{d\left( {{\tan }^{-1}}x \right)}{dx}=\dfrac{1}{1+{{x}^{2}}} $ . Once you get the derivative report the coefficient of $ \sec \left( {{\tan }^{-1}}x \right) $ as your answer. You may have to use the identity $ \tan \left( {{\tan }^{-1}}x \right)=x $ for simplification of your answer.
Complete step-by-step answer:
Let us start with finding the derivative of $ \sec \left( {{\tan }^{-1}}x \right) $ . For finding the derivative, we will use the chain rule of differentiation. According to the chain rule of differentiation, we know $ \dfrac{d\left( f\left( g(x) \right) \right)}{dx}=f'\left( g(x) \right)\times g'(x) $ . For $ \sec \left( {{\tan }^{-1}}x \right) $ , $ f\left( x \right)=\sec x $ and $ g(x)={{\tan }^{-1}}x $ .
We also know that $ \dfrac{d\left( \sec x \right)}{dx}=\sec x\tan x $ . .
$ \therefore \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\tan \left( {{\tan }^{-1}}x \right)\times \dfrac{d\left( {{\tan }^{-1}}x \right)}{dx} $
Now, we know that $ {{\tan }^{-1}}x=\dfrac{1}{1+{{x}^{2}}} $ . So, if we use this in our equation, we get
$ \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\tan \left( {{\tan }^{-1}}x \right)\times \dfrac{1}{1+{{x}^{2}}} $
Now, according to the rules of inverse trigonometric function, we know $ \tan \left( {{\tan }^{-1}}x \right)=x $ . If we use this in our equation, we get
$ \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\times x\times \dfrac{1}{1+{{x}^{2}}} $
$ \Rightarrow \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\times \dfrac{x}{1+{{x}^{2}}} $
Now, we are asked the differential coefficient of $ \sec \left( {{\tan }^{-1}}x \right) $ . So, from the above result, we can clearly see that the coefficient of $ \sec \left( {{\tan }^{-1}}x \right) $ in the derivative of $ \sec \left( {{\tan }^{-1}}x \right) $ is $ \dfrac{x}{1+{{x}^{2}}} $ .
So, the correct answer is “Option A”.
Note: The first thing to keep in mind is that according to the rule of trigonometric inverse functions, $ \tan \left( {{\tan }^{-1}}x \right)=x $ , but $ {{\tan }^{-1}}\left( \tan x \right)=x $ if and only if it is mentioned that $ x\in \left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right) $ . The other thing to keep in mind is while you report the answers, report the exact matched option, as the options given are very similar and misleading.
Complete step-by-step answer:
Let us start with finding the derivative of $ \sec \left( {{\tan }^{-1}}x \right) $ . For finding the derivative, we will use the chain rule of differentiation. According to the chain rule of differentiation, we know $ \dfrac{d\left( f\left( g(x) \right) \right)}{dx}=f'\left( g(x) \right)\times g'(x) $ . For $ \sec \left( {{\tan }^{-1}}x \right) $ , $ f\left( x \right)=\sec x $ and $ g(x)={{\tan }^{-1}}x $ .
We also know that $ \dfrac{d\left( \sec x \right)}{dx}=\sec x\tan x $ . .
$ \therefore \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\tan \left( {{\tan }^{-1}}x \right)\times \dfrac{d\left( {{\tan }^{-1}}x \right)}{dx} $
Now, we know that $ {{\tan }^{-1}}x=\dfrac{1}{1+{{x}^{2}}} $ . So, if we use this in our equation, we get
$ \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\tan \left( {{\tan }^{-1}}x \right)\times \dfrac{1}{1+{{x}^{2}}} $
Now, according to the rules of inverse trigonometric function, we know $ \tan \left( {{\tan }^{-1}}x \right)=x $ . If we use this in our equation, we get
$ \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\times x\times \dfrac{1}{1+{{x}^{2}}} $
$ \Rightarrow \dfrac{d\left( \sec \left( {{\tan }^{-1}}x \right) \right)}{dx}=\sec \left( {{\tan }^{-1}}x \right)\times \dfrac{x}{1+{{x}^{2}}} $
Now, we are asked the differential coefficient of $ \sec \left( {{\tan }^{-1}}x \right) $ . So, from the above result, we can clearly see that the coefficient of $ \sec \left( {{\tan }^{-1}}x \right) $ in the derivative of $ \sec \left( {{\tan }^{-1}}x \right) $ is $ \dfrac{x}{1+{{x}^{2}}} $ .
So, the correct answer is “Option A”.
Note: The first thing to keep in mind is that according to the rule of trigonometric inverse functions, $ \tan \left( {{\tan }^{-1}}x \right)=x $ , but $ {{\tan }^{-1}}\left( \tan x \right)=x $ if and only if it is mentioned that $ x\in \left( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right) $ . The other thing to keep in mind is while you report the answers, report the exact matched option, as the options given are very similar and misleading.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english 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)
Write a letter to the principal requesting him to grant class 10 english CBSE
![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)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths 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)
Difference Between Plant Cell and Animal Cell
![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)