
The value of \[\dfrac{d}{{dx}}{\cosh ^{ - 1}}(\sec x)\] is equal to:
A. \[\sec x\]
B. \[\tan x\]
C. \[\sin x\]
D. \[{\mathop{\rm cosec}\nolimits} x\]
Answer
164.4k+ views
Hint:
The given question will be solved by a chain rule method. First we will write the differentiation of \[\cosh ^{ - 1}x \] where \[x =\sec x\]. Then multiply the differentiation of \[\sec x\].
Formula Used:
\[\dfrac{d}{{dx}}{\cosh ^{ - 1}}x = \dfrac{1}{{\sqrt {{x^2} - 1} }}\]
\[\dfrac{d}{{dx}}{\sec x = {\sec x\tan x}}\]
\[{\tan ^2}{\rm{ x = }}{\sec ^2}x - 1\]
Complete step-by-step answer:
We have been given the function \[\dfrac{d}{{dx}}{\cosh ^{ - 1}}(\sec x)\].
Let \[y = {\cosh ^{ - 1}}(\sec x)\]
Differentiating y with respect to x, we get,
\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{\sqrt {\left( {{{\sec }^2}x - 1} \right)} }}\sec x\tan x\]
Now we will apply the property of \[{\tan ^2}{\rm{ x = }}{\sec ^2}x - 1\]
\[ = \dfrac{{\sec x\tan x}}{{\sqrt {{{\tan }^2}x} }}\]
Now take the square root of \[{\tan ^2}x\]
\[ = \dfrac{{\sec x\tan x}}{{\tan x}}\]
\[ = \sec x\]
Hence, option A is correct.
Additional information
Chain rule is used to differentiate composite functions. The differentiate of a composite
\[f\left( {g\left( x \right)} \right)\] is \[\frac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right)g'\left( x \right)\]
To apply chain rule, we need to identify the outer function and inner function. Then we will write the derivative of outer function and multiply the differentiation of inner function.
Note:
Students often make common mistakes to find the differentiation of \[\frac{d}{{dx}}{\cosh ^{ - 1}}(\sec x)\]. They do not use chain rule to solve the question and treat \[\sec x\] as \[x\]. The correct way is first to differentiate \[{\cosh ^{ - 1}}(\sec x)\] and multiply the differentiation of \[\sec x\] with it.
The given question will be solved by a chain rule method. First we will write the differentiation of \[\cosh ^{ - 1}x \] where \[x =\sec x\]. Then multiply the differentiation of \[\sec x\].
Formula Used:
\[\dfrac{d}{{dx}}{\cosh ^{ - 1}}x = \dfrac{1}{{\sqrt {{x^2} - 1} }}\]
\[\dfrac{d}{{dx}}{\sec x = {\sec x\tan x}}\]
\[{\tan ^2}{\rm{ x = }}{\sec ^2}x - 1\]
Complete step-by-step answer:
We have been given the function \[\dfrac{d}{{dx}}{\cosh ^{ - 1}}(\sec x)\].
Let \[y = {\cosh ^{ - 1}}(\sec x)\]
Differentiating y with respect to x, we get,
\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{\sqrt {\left( {{{\sec }^2}x - 1} \right)} }}\sec x\tan x\]
Now we will apply the property of \[{\tan ^2}{\rm{ x = }}{\sec ^2}x - 1\]
\[ = \dfrac{{\sec x\tan x}}{{\sqrt {{{\tan }^2}x} }}\]
Now take the square root of \[{\tan ^2}x\]
\[ = \dfrac{{\sec x\tan x}}{{\tan x}}\]
\[ = \sec x\]
Hence, option A is correct.
Additional information
Chain rule is used to differentiate composite functions. The differentiate of a composite
\[f\left( {g\left( x \right)} \right)\] is \[\frac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right)g'\left( x \right)\]
To apply chain rule, we need to identify the outer function and inner function. Then we will write the derivative of outer function and multiply the differentiation of inner function.
Note:
Students often make common mistakes to find the differentiation of \[\frac{d}{{dx}}{\cosh ^{ - 1}}(\sec x)\]. They do not use chain rule to solve the question and treat \[\sec x\] as \[x\]. The correct way is first to differentiate \[{\cosh ^{ - 1}}(\sec x)\] and multiply the differentiation of \[\sec x\] with it.
Recently Updated Pages
Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

JEE Electricity and Magnetism Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics

JEE Advanced 2025 Notes
