
If \[\varphi \left( x \right)\] is the inverse of the function f(x) and \[f^{'}\left( x \right) = \dfrac{1}{{\left( {1 + {x^5}} \right)}}\] then what is the value of \[\dfrac{d}{{dx}}\varphi \left( x \right)\]?
A.\[\dfrac{1}{{\left( {1 + {{\left( {\varphi \left( 5 \right)} \right)}^5}} \right)}}\]
B.\[1 + f\left( x \right)\]
C.\[\left( {1 + {{\left( {\varphi \left( 5 \right)} \right)}^5}} \right)\]
D.\[\dfrac{1}{{\left( {1 + {{\left( {f\left( 5 \right)} \right)}^5}} \right)}}\]
Answer
164.1k+ views
Hint: We know that composite of a function and inverse return \[x\]. Apply this condition to establish a relation between \[f\left( x \right)\] and \[\phi \left( x \right)\]. Then we will derivative \[fo\varphi \left( x \right) = x\] with respect to \[x\] using chain rule. After that divide both sides by \[f'\left( {\phi \left( x \right)} \right)\]. Then find the value of \[f'\left( {\phi \left( x \right)} \right)\] by putting \[x = \phi \left( x \right)\] in \[f^{'}\left( x \right) = \dfrac{1}{{\left( {1 + {x^5}} \right)}}\]. Then we will get desire result.
Formula Used:
We will use the composite function i.e., \[f\left( {g\left( x \right)} \right)\], and derivative formula \[\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f^{'}\left( {g\left( x \right)} \right) \times g^{'}x\] and \[\dfrac{d}{{dx}}x = 1\] and by using chain rule.
Complete step by step solution:
Given \[f^{'}\left( x \right) = \dfrac{1}{{\left( {1 + {x^5}} \right)}}\], and \[\varphi \left( x \right)\] is the inverse of the function f(x),
\[ \Rightarrow f\left( x \right) = \varphi^{\prime} \left( x \right)\],
Now we will use the composite function rule, we will get,
\[ \Rightarrow \varphi \left( x \right) = x\]
Now we will differentiate on both sides we get,
\[ \Rightarrow \dfrac{d}{{dx}}f o \varphi \left( x \right) = \dfrac{d}{{dx}}x\]
\[ \Rightarrow f'\left( \varphi(x) \right) . \varphi^{\prime} (x) = 1\]
\[ \Rightarrow \dfrac{1}{{\left( {1 + {x^5}} \right)}} . \varphi^{\prime} (x) = 1\]
\[ \Rightarrow \varphi^{\prime} (x) = {1 + {(x)^5}}\]
The correct option is C.
Note: Students often do a common thing, that is they apply product rules on \[f \circ g\left( x \right)\]. They calculate the derivative \[f \circ g\left( x \right)\] as \[f'\left( x \right)g\left( x \right) + f\left( x \right)g'\left( x \right)\]. That is an incorrect way . The correct way is \[\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\].
Formula Used:
We will use the composite function i.e., \[f\left( {g\left( x \right)} \right)\], and derivative formula \[\dfrac{d}{{dx}}f\left( {g\left( x \right)} \right) = f^{'}\left( {g\left( x \right)} \right) \times g^{'}x\] and \[\dfrac{d}{{dx}}x = 1\] and by using chain rule.
Complete step by step solution:
Given \[f^{'}\left( x \right) = \dfrac{1}{{\left( {1 + {x^5}} \right)}}\], and \[\varphi \left( x \right)\] is the inverse of the function f(x),
\[ \Rightarrow f\left( x \right) = \varphi^{\prime} \left( x \right)\],
Now we will use the composite function rule, we will get,
\[ \Rightarrow \varphi \left( x \right) = x\]
Now we will differentiate on both sides we get,
\[ \Rightarrow \dfrac{d}{{dx}}f o \varphi \left( x \right) = \dfrac{d}{{dx}}x\]
\[ \Rightarrow f'\left( \varphi(x) \right) . \varphi^{\prime} (x) = 1\]
\[ \Rightarrow \dfrac{1}{{\left( {1 + {x^5}} \right)}} . \varphi^{\prime} (x) = 1\]
\[ \Rightarrow \varphi^{\prime} (x) = {1 + {(x)^5}}\]
The correct option is C.
Note: Students often do a common thing, that is they apply product rules on \[f \circ g\left( x \right)\]. They calculate the derivative \[f \circ g\left( x \right)\] as \[f'\left( x \right)g\left( x \right) + f\left( x \right)g'\left( x \right)\]. That is an incorrect way . The correct way is \[\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\].
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 Main 2025 Session 2: Exam Date, Admit Card, Syllabus, & More

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

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

Trending doubts
Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main Chemistry Question Paper with Answer Keys and Solutions

JEE Main Reservation Criteria 2025: SC, ST, EWS, and PwD Candidates

What is Normality in Chemistry?

Chemistry Electronic Configuration of D Block Elements: JEE Main 2025

Other Pages
Total MBBS Seats in India 2025: Government College Seat Matrix

NEET Total Marks 2025: Important Information and Key Updates

Neet Cut Off 2025 for MBBS in Tamilnadu: AIQ & State Quota Analysis

Karnataka NEET Cut off 2025 - Category Wise Cut Off Marks

NEET Marks vs Rank 2024|How to Calculate?

NEET 2025: All Major Changes in Application Process, Pattern and More
