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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?
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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.
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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?
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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?.
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