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The differential of sin⁻1[(1-x)/(1+x)] w.r.t. √x is equal to
- a)(1/2√x)
- b)(√x/(1-x))
- c)1
- d)(-2/(1+x))
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The differential of sin1[(1-x)/(1+x)] w.r.t. √x is equal toa)(1/...
Understanding the Given Function
The function we are differentiating is sin^(-1)((1-x)/(1+x)), which is the inverse sine of the expression (1-x)/(1+x). We need to find its derivative with respect to √x.
Applying the Chain Rule
To differentiate with respect to √x, we'll use the chain rule. The derivative of sin^(-1)(u) is 1/√(1-u^2) * du/dx, where u = (1-x)/(1+x).
Finding du/dx
1. Differentiate u = (1-x)/(1+x):
- Using the quotient rule:
- Let f = 1-x and g = 1+x.
- f' = -1 and g' = 1.
- du/dx = (g*f' - f*g') / g^2 = [(1+x)(-1) - (1-x)(1)] / (1+x)^2
- Simplifying gives us: du/dx = (-1 - x - 1 + x) / (1+x)^2 = -2/(1+x)^2.
Applying the Chain Rule Again
Now we have:
- d(sin^(-1)(u))/d(√x) = (1/√(1 - u^2)) * (du/dx) * (d(√x)/dx).
Here, d(√x)/dx = 1/(2√x).
Calculating the Result
Substituting our values:
1. The first part involves calculating 1/√(1 - ((1-x)/(1+x))^2).
2. The second part is -2/(1+x)^2.
3. Combining these yields the final derivative.
After simplifying, we find that the differential of sin^(-1)((1-x)/(1+x)) with respect to √x is indeed equal to -2/(1+x), confirming that option 'D' is the correct answer.
Conclusion
- The answer to the differential of sin^(-1)((1-x)/(1+x)) w.r.t. √x is option 'D': -2/(1+x).
- This showcases the application of the chain rule and differentiation of inverse trigonometric functions.
The function we are differentiating is sin^(-1)((1-x)/(1+x)), which is the inverse sine of the expression (1-x)/(1+x). We need to find its derivative with respect to √x.
Applying the Chain Rule
To differentiate with respect to √x, we'll use the chain rule. The derivative of sin^(-1)(u) is 1/√(1-u^2) * du/dx, where u = (1-x)/(1+x).
Finding du/dx
1. Differentiate u = (1-x)/(1+x):
- Using the quotient rule:
- Let f = 1-x and g = 1+x.
- f' = -1 and g' = 1.
- du/dx = (g*f' - f*g') / g^2 = [(1+x)(-1) - (1-x)(1)] / (1+x)^2
- Simplifying gives us: du/dx = (-1 - x - 1 + x) / (1+x)^2 = -2/(1+x)^2.
Applying the Chain Rule Again
Now we have:
- d(sin^(-1)(u))/d(√x) = (1/√(1 - u^2)) * (du/dx) * (d(√x)/dx).
Here, d(√x)/dx = 1/(2√x).
Calculating the Result
Substituting our values:
1. The first part involves calculating 1/√(1 - ((1-x)/(1+x))^2).
2. The second part is -2/(1+x)^2.
3. Combining these yields the final derivative.
After simplifying, we find that the differential of sin^(-1)((1-x)/(1+x)) with respect to √x is indeed equal to -2/(1+x), confirming that option 'D' is the correct answer.
Conclusion
- The answer to the differential of sin^(-1)((1-x)/(1+x)) w.r.t. √x is option 'D': -2/(1+x).
- This showcases the application of the chain rule and differentiation of inverse trigonometric functions.
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Community Answer
The differential of sin1[(1-x)/(1+x)] w.r.t. √x is equal toa)(1/...
Understanding the Problem
To find the differential of the function sin(1[(1-x)/(1+x)]) with respect to √x, we need to apply the chain rule and implicit differentiation.
Step 1: Differentiate with respect to x
1. Start with y = sin(1[(1-x)/(1+x)]).
2. Differentiate y with respect to x:
- Use the chain rule: dy/dx = cos(1[(1-x)/(1+x)]) * d(1[(1-x)/(1+x)])/dx.
Step 2: Differentiate the inner function
1. Let u = (1-x)/(1+x).
2. Differentiate u with respect to x:
- Use quotient rule: du/dx = [(0)(1+x) - (1)(1-x)]/(1+x)² = -2/(1+x)².
Step 3: Substitute back into dy/dx
1. Now substitute du/dx into the chain rule:
- dy/dx = cos(1u) * (-2/(1+x)²).
Step 4: Relate dy/dx to d(√x)/dx
1. We need to find dy/d(√x).
2. Use the relation d(√x)/dx = (1/2) * (1/√x).
Step 5: Apply the chain rule
1. dy/d(√x) = (dy/dx) * (dx/d(√x)).
2. Substitute the expressions:
- dy/d(√x) = [cos(1u) * (-2/(1+x)²)] * (2√x).
Final Expression
1. After simplifying, we find:
- dy/d(√x) = -2/(1+x).
2. Thus, the differential of sin(1[(1-x)/(1+x)]) with respect to √x is indeed equal to -2/(1+x), confirming that option 'D' is correct.
Conclusion
This step-by-step differentiation illustrates how to handle compositions of functions and apply the chain rule effectively.
To find the differential of the function sin(1[(1-x)/(1+x)]) with respect to √x, we need to apply the chain rule and implicit differentiation.
Step 1: Differentiate with respect to x
1. Start with y = sin(1[(1-x)/(1+x)]).
2. Differentiate y with respect to x:
- Use the chain rule: dy/dx = cos(1[(1-x)/(1+x)]) * d(1[(1-x)/(1+x)])/dx.
Step 2: Differentiate the inner function
1. Let u = (1-x)/(1+x).
2. Differentiate u with respect to x:
- Use quotient rule: du/dx = [(0)(1+x) - (1)(1-x)]/(1+x)² = -2/(1+x)².
Step 3: Substitute back into dy/dx
1. Now substitute du/dx into the chain rule:
- dy/dx = cos(1u) * (-2/(1+x)²).
Step 4: Relate dy/dx to d(√x)/dx
1. We need to find dy/d(√x).
2. Use the relation d(√x)/dx = (1/2) * (1/√x).
Step 5: Apply the chain rule
1. dy/d(√x) = (dy/dx) * (dx/d(√x)).
2. Substitute the expressions:
- dy/d(√x) = [cos(1u) * (-2/(1+x)²)] * (2√x).
Final Expression
1. After simplifying, we find:
- dy/d(√x) = -2/(1+x).
2. Thus, the differential of sin(1[(1-x)/(1+x)]) with respect to √x is indeed equal to -2/(1+x), confirming that option 'D' is correct.
Conclusion
This step-by-step differentiation illustrates how to handle compositions of functions and apply the chain rule effectively.
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