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What will be the differential function of √(x2 + 2)?
- a)x√(x2 + 2) dx
- b)x/√(x2 + 2) dx
- c)x/√(x2 – 2) dx
- d)-x/√(x2 + 2) dx
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
What will be the differential function of √(x2 + 2)?a)x√(x...
Let, y = f(x) = √(x2 + 2)
So, f(x) = (x2 + 2)1/2
On differentiating it we get,
f’(x) = d/dx[(x2 + 2)1/2]
f’(x) = 1/2 * 1/√(x2 + 2) * 2x
So f’(x) = x/√(x2 + 2)
So the differential equation is:
dy = f’(x)dx
Hence, dy = x/√(x2 + 2) dx
So, f(x) = (x2 + 2)1/2
On differentiating it we get,
f’(x) = d/dx[(x2 + 2)1/2]
f’(x) = 1/2 * 1/√(x2 + 2) * 2x
So f’(x) = x/√(x2 + 2)
So the differential equation is:
dy = f’(x)dx
Hence, dy = x/√(x2 + 2) dx
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Community Answer
What will be the differential function of √(x2 + 2)?a)x√(x...
Understanding the Problem
To find the derivative of the function f(x) = √(x² + 2), we will apply the rules of differentiation. We want to find f'(x).
Applying the Chain Rule
1. The function can be rewritten as f(x) = (x² + 2)^(1/2).
2. We will use the chain rule which states that if you have a composite function g(h(x)), then the derivative is g'(h(x)) * h'(x).
Identifying Components
- Let u = x² + 2.
- Therefore, f(x) = u^(1/2).
Finding Derivatives
- Step 1: Differentiate u with respect to x:
- u' = d/dx (x² + 2) = 2x.
- Step 2: Differentiate f with respect to u:
- f'(u) = d/du (u^(1/2)) = (1/2)u^(-1/2).
Combining Derivatives
Now, using the chain rule:
- f'(x) = f'(u) * u' = (1/2)(u^(-1/2))(2x).
Substituting u back:
- f'(x) = (1/2)(x² + 2)^(-1/2) * (2x) = x / √(x² + 2).
Conclusion
The correct differential function of √(x² + 2) is indeed:
- Option B: x / √(x² + 2) dx.
This is the derivative we sought, confirming that option B is the correct answer.
To find the derivative of the function f(x) = √(x² + 2), we will apply the rules of differentiation. We want to find f'(x).
Applying the Chain Rule
1. The function can be rewritten as f(x) = (x² + 2)^(1/2).
2. We will use the chain rule which states that if you have a composite function g(h(x)), then the derivative is g'(h(x)) * h'(x).
Identifying Components
- Let u = x² + 2.
- Therefore, f(x) = u^(1/2).
Finding Derivatives
- Step 1: Differentiate u with respect to x:
- u' = d/dx (x² + 2) = 2x.
- Step 2: Differentiate f with respect to u:
- f'(u) = d/du (u^(1/2)) = (1/2)u^(-1/2).
Combining Derivatives
Now, using the chain rule:
- f'(x) = f'(u) * u' = (1/2)(u^(-1/2))(2x).
Substituting u back:
- f'(x) = (1/2)(x² + 2)^(-1/2) * (2x) = x / √(x² + 2).
Conclusion
The correct differential function of √(x² + 2) is indeed:
- Option B: x / √(x² + 2) dx.
This is the derivative we sought, confirming that option B is the correct answer.
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What will be the differential function of √(x2 + 2)?a)x√(x2 + 2) dxb)x/√(x2 + 2) dxc)x/√(x2 – 2) dxd)-x/√(x2 + 2) dxCorrect answer is option 'B'. Can you explain this answer?
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