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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.
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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?
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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?.
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