UPSC Exam > UPSC Questions > Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ...
Start Learning for Free
Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is
2/13
(9) 1/2
(c) 11/13
IIT JAM-2014
(d) 11/4?
2/13
(9) 1/2
(c) 11/13
IIT JAM-2014
(d) 11/4?
Most Upvoted Answer
Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R I...
Understanding the Functions
The given functions are:
- f(x) = x^3 + x
- g(x) = x^3 - x
We need to find the derivative of the composite function g(f^(-1)(x)) at the point x = 2.
Finding the Inverse Function f^(-1)(x)
To find f^(-1)(2), we solve f(x) = 2:
1. Start with the equation: x^3 + x = 2.
2. Rearranging gives us: x^3 + x - 2 = 0.
Using numerical methods or graphing, we can find that one root is approximately x = 1.
Thus, f^(-1)(2) = 1.
Applying the Chain Rule
Next, we need to find the derivative of g(f^(-1)(x)) using the chain rule:
1. g(f^(-1)(x))' = g'(f^(-1)(x)) * f^(-1)'(x).
Calculating g'(x)
g'(x) = 3x^2 - 1.
Now, substituting f^(-1)(2):
1. Calculate g'(1): g'(1) = 3(1)^2 - 1 = 2.
Finding f'(x)
To find f'(x):
f'(x) = 3x^2 + 1.
Now, calculate f'(1):
1. f'(1) = 3(1)^2 + 1 = 4.
Finding f^(-1)'(2)
By the inverse function theorem:
f^(-1)'(2) = 1 / f'(f^(-1)(2)) = 1 / f'(1) = 1 / 4.
Putting It All Together
Now we can find g(f^(-1)(2))':
1. g(f^(-1)(2))' = g'(1) * f^(-1)'(2) = 2 * (1/4) = 1/2.
Thus, the derivative of the composite function g(f^(-1)(x)) at x = 2 is 1/2.
The given functions are:
- f(x) = x^3 + x
- g(x) = x^3 - x
We need to find the derivative of the composite function g(f^(-1)(x)) at the point x = 2.
Finding the Inverse Function f^(-1)(x)
To find f^(-1)(2), we solve f(x) = 2:
1. Start with the equation: x^3 + x = 2.
2. Rearranging gives us: x^3 + x - 2 = 0.
Using numerical methods or graphing, we can find that one root is approximately x = 1.
Thus, f^(-1)(2) = 1.
Applying the Chain Rule
Next, we need to find the derivative of g(f^(-1)(x)) using the chain rule:
1. g(f^(-1)(x))' = g'(f^(-1)(x)) * f^(-1)'(x).
Calculating g'(x)
g'(x) = 3x^2 - 1.
Now, substituting f^(-1)(2):
1. Calculate g'(1): g'(1) = 3(1)^2 - 1 = 2.
Finding f'(x)
To find f'(x):
f'(x) = 3x^2 + 1.
Now, calculate f'(1):
1. f'(1) = 3(1)^2 + 1 = 4.
Finding f^(-1)'(2)
By the inverse function theorem:
f^(-1)'(2) = 1 / f'(f^(-1)(2)) = 1 / f'(1) = 1 / 4.
Putting It All Together
Now we can find g(f^(-1)(2))':
1. g(f^(-1)(2))' = g'(1) * f^(-1)'(2) = 2 * (1/4) = 1/2.
Thus, the derivative of the composite function g(f^(-1)(x)) at x = 2 is 1/2.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4?
Question Description
Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4?.
Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4?.
Solutions for Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? defined & explained in the simplest way possible. Besides giving the explanation of
Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4?, a detailed solution for Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? has been provided alongside types of Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? theory, EduRev gives you an
ample number of questions to practice Let f(x) = x ^ 3 + x and \mathfrak{g}(x) = x ^ 3 - x for all x \in R If f¹ denotes the inverse function of f, then the derivative of the composite go f' at the point 2 is2/13(9) 1/2(c) 11/13IIT JAM-2014(d) 11/4? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup