Mathematics Exam > Mathematics Questions > Let G:(0,inf)-->R be a differentiable functio...
Start Learning for Free
Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals?
Most Upvoted Answer
Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^...
Given:
- A differentiable function G:(0,inf)-->R
- f(x^2)= 1-x^3 for all x>0
- f(1) = 0
To find:
- The value of f(4)
Solution:
Step 1: Finding the derivative of f(x)
To find f'(x), we need to differentiate both sides of the given equation f(x^2) = 1 - x^3 with respect to x.
Differentiating the left side of the equation using the chain rule, we get:
f'(x^2) * d/dx(x^2) = -3x^2
Simplifying this, we have:
2x * f'(x^2) = -3x^2
Dividing both sides by 2x, we get:
f'(x^2) = -3x/2
Step 2: Finding the derivative of f(x^2) with respect to x
To find the derivative of f(x^2) with respect to x, we need to apply the chain rule.
Letting u = x^2, we have:
f'(u) * d/du(u) = f'(u) * d/dx(x^2) = f'(u) * 2x
Substituting f'(u) = -3x/2 from Step 1, we get:
-3x/2 * 2x = -3x^2
Step 3: Finding f'(x)
Since f(x^2) = 1 - x^3, we can express f(x) as:
f(x) = f(x^2) = 1 - x^3
Differentiating both sides of this equation with respect to x, we get:
f'(x) = -3x^2
Step 4: Finding f(4)
To find f(4), we substitute x = 4 into the equation f(x) = 1 - x^3.
f(4) = 1 - 4^3
f(4) = 1 - 64
f(4) = -63
Therefore, f(4) equals -63.
- A differentiable function G:(0,inf)-->R
- f(x^2)= 1-x^3 for all x>0
- f(1) = 0
To find:
- The value of f(4)
Solution:
Step 1: Finding the derivative of f(x)
To find f'(x), we need to differentiate both sides of the given equation f(x^2) = 1 - x^3 with respect to x.
Differentiating the left side of the equation using the chain rule, we get:
f'(x^2) * d/dx(x^2) = -3x^2
Simplifying this, we have:
2x * f'(x^2) = -3x^2
Dividing both sides by 2x, we get:
f'(x^2) = -3x/2
Step 2: Finding the derivative of f(x^2) with respect to x
To find the derivative of f(x^2) with respect to x, we need to apply the chain rule.
Letting u = x^2, we have:
f'(u) * d/du(u) = f'(u) * d/dx(x^2) = f'(u) * 2x
Substituting f'(u) = -3x/2 from Step 1, we get:
-3x/2 * 2x = -3x^2
Step 3: Finding f'(x)
Since f(x^2) = 1 - x^3, we can express f(x) as:
f(x) = f(x^2) = 1 - x^3
Differentiating both sides of this equation with respect to x, we get:
f'(x) = -3x^2
Step 4: Finding f(4)
To find f(4), we substitute x = 4 into the equation f(x) = 1 - x^3.
f(4) = 1 - 4^3
f(4) = 1 - 64
f(4) = -63
Therefore, f(4) equals -63.
|
Explore Courses for Mathematics exam
|
|
Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals?
Question Description
Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals?.
Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals?.
Solutions for Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? defined & explained in the simplest way possible. Besides giving the explanation of
Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals?, a detailed solution for Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? has been provided alongside types of Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? theory, EduRev gives you an
ample number of questions to practice Let G:(0,inf)-->R be a differentiable function such that f'(x^2)= 1-x^3 for all x>0 and f(1) =0 then f(4) equals? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Suggested Free Tests
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